I think I can answer this one, and it's actually quite easy to do depending on just how accurate you want to get your tail stock aligned and doesn't require anything too fancy. Lazers are really inaccurate and can't measure anything unless your using the multi multi 10's of thousands
$$$$ of very specialized lazer systems that can measure extremely accurately. But I said easy, not short to explain because the proper answer just isn't that simple. And there's various levels of experiece any may or may not have that might read this later.
You would need at a minimum a dial or digital caliper, for multiple good reasons and if I was buying my first set of calipers, I'd not be buying at the cheapest possible price. More in the mid range price level. $60 level or a bit more. Better of course would be an actual micrometer. But I need to start with some basic measurement terms, methods and accuracy definitions first. And contrary to what most assume including most metal working hobbyist's, dial or digital calipers are only trustworthy to at best + - .002" for even the very best one's. If it's important or requires being exact enough, I don't trust any caliper measurement including my expensive Mitutoyo's to be completely trustworthy under about .005". And yes I have checked mine and my own measurement techniques against proper and certified gauge blocks. Yes they will do much better than that .005", but there best accuracy is only under ideal conditions and being very careful about my technique. When I need to know it's correct and actually be sure it's under that .005" number, then caliper measurements aren't the correct tool. Used within there limits they are a fast and excellent tool, and for my own use, they get me close enough to where I might then need better and much higher accuracy measurements. Or sometimes caliper measurements are more than good enough. For checking that tail stock alignment, calipers could still get you pretty close, or close enough.
However and it literally doesn't matter what the dial or display is showing, the mechanical assembly and design calipers use isn't really capable of much better than that .002". What the dial or display divisions are showing on any measurement tool is it's resolution and absolutely not the tools level of accuracy. Almost everyone keeps getting that confused or misunderstood. The resolution term needs to be properly understood and it only means how many digits there are to the right of the decimal point. Or the tools lowest division it's capable of displaying as an analog or digital number. A measuring tools repeatability and accuracy are both something completely different. A measurement tools repeatability also doesn't mean it's guaranteed to be highly accurate. It can be extremely repeatable over multiple measurements, and that's also highly desirable, but if it's always for example .005" from the actual part size, it's still not an accurate measurement. However when used as a comparison gauge or tool, calipers can still be used quite accurately "IF" good repeatable measurement techniques are used. For smaller diameter items, it's best to slowly and lightly close the calipers jaws using your thumb and forefinger and not that thumb wheel to gently close the jaws while wiggling them a bit to ensure the calipers jaw faces are seating down flat and square onto the part surface. For larger measurements where you can't use your thumb and fore finger, even Mitutoyo recomends gently closing the caliper against the part using both hands for the best accuracy any caliper can deliver. That thumb wheel most calipers have is only really meant for fast coarse adjustments. It can be used while measuring, as long as you understand it takes such a light and gentle touch first, and that takes a lot of practice. Always use as close as possible the same amount of closing pressure between your thumb and forefinger, or closing them with both hands should you decide to use that method.
Secondly and when ever possible do NOT measure out towards the jaw tips with any caliper. Because of their mechanical design, measuring out towards the jaw tips just increases inaccuracy. Sometimes there's no choice, but it does increase the chances of even less accurate measurements. The moving jaw due to very minor but still mandatory clearances to allow it to move along the beam of the caliper also allow that jaw to flex a tiny amount either inwards or outwards. The jaw faces then aren't properly parallel to each other. On round surfaces, that's even more important to measure as deep inside the jaws as possible. The inside caliper jaws and measuring hole sizes are even less accurate for a few very good reasons I won't go into right now. Ok all this is maybe a bit more than most here might want to know about measurement. But it's important enough to matter, and there's simply way too much misunderstood, and while almost for sure well meant, still completely wrong information about accurate caliper measurements almost everywhere online. My Mitutoyo calipers have a .0005" digit on the end of its display. That was added probably because it was cheap to do, and some might even believe it enough to be an increased sales tactic. But I never pay any attention to that number because the calipers simply can't measure down to that level other than just randomly being lucky once in awhile. With my own checking against randomly picked gauge block sizes, that might only happen 1-3 times out of 10 measurements.
There's a lot more to lathe alignment, but as far as getting the tail stocks center line centered on the head stocks center line. I'm trying to keep this simple and about what was asked. First measure the tail stocks quill outside diameter. Again that consistent measurement technique. And I'd want to take at least 5 measurements to double check there isn't any inconsistency in your measurements. Write each one down and average them if needed. Now it's probably a bit tricky with hand turning, but a highly experienced turner should be able to do it even if it takes a few trys at it. Chuck up something like a piece of Delrin plastic, or likely harder to do well, a piece of 6061 aluminum that's a bit larger in diameter that your tail stock quills outside diameter. Or I suppose a short scrap of extremely dense and very hard hardwood could be used instead. It would need to be a wood type that doesn't compress easily at all. And you only need a short piece of whatever material your turning, at most maybe an inch or two long outside the chuck jaws. From now on that piece stays in the chuck until the test measurements are completely done. By however you want to do it and I'm not a wood turner yet, face the part flat or very slightly concave, then turn a short parallel section maybe 1/2" long to the same diameter as your tail stocks quill measurement is. Take as long as it takes to get that correct and as exact as possible. Very fine sand paper at the end would likely getting to a matching size and parallel easier, and you also want a very smooth and even surface finish. It also wouldn't hurt to comparison measure the diameter your attempting to turn and sand down, and the tail stock quill OD as continuing trial measurements as you slowly creep up on getting that matching dimension.
Ok lets assume you've now got that turned part end to an exact size done and your turned and sanded parts OD does match the tail stocks OD. Now spotlessly clean the lathe bed, remove the tool rest and slide the tail stock up until the face of it's quill almost but not quite touches the face of your turned part. With a wood turning lathe and no carriage in the way like a metal turning lathe would have, there's two separate ways I'd take my tail stock measurements. The first with the tail stocks quill retracted, but always with the bed and quill lock set. The second, move the tail stock back a bit and extend the quill almost to it's full travel extension. Again set the bed and quill lock before any measurements are taken. That's important! But at no time do you want to ever bump against the face of your turned part. A good way to gauge where the end of your tail stocks quill is would be a strip of printer paper between the face of your part and the face of the tail stocks quill. Slowly keep extending the quill until you just start feel a light drag on the paper as it's lightly pulled past both surfaces that are facing each other. That should get the end of the tail stocks quill within around .003"- .004" from touching the face of your turned part. Again set the tail stocks bed and quill lock.
Now set your caliper jaws so each side of each jaw spans across both your turned part and the quills OD. Visually keep your caliper or micrometer square and 90 degrees to the part and tail stocks quill. Measure with the caliper or micrometer held vertically. Again I'd take at least 5 separate measurements just to be sure and write each one down. Now measure across those same two surfaces with the caliper or micrometer held in the horizontal position, again at least 5 measurements and written down. Any tail stock misalignment either vertically or horizontally will show a larger measurement number in each direction than your tail stocks quill OD is. It also measures how much the tail stocks quill is out to the lathes head stock. Make your adjustments, remeasure etc, etc, etc until you get the alignment correct. Only then can you remove your turned part from the chuck.
There is a slightly less involved, quicker and a bit less accurate method using two steel dead centers in the head and tail stock and a steel ruler between them. But it's best used with metal turning lathes where a fresh 60 degree center point can be turned on a piece of scrap metal so it's known to be 100% concentric to the head stocks center of rotation.
Outside the original thread question, but just in case and for any that might have one, a dial indicator and magnetic base attached to either the lathe chuck or face plate and rotating that around your tail stocks quill WILL NOT WORK!!!! And this short video shows precisely why.
View: https://www.youtube.com/watch?v=u08SfxVgxNg
Decades before that video was put up, I spent a couple of very frustrating hrs trying to figure out why my tail stocks center line appeared to be quite low compared to my head stock before I finally clued in what the actual cause was while swinging an indicator around the tail stocks OD. But even for wood workers and especially those with wood surfacing, cutting and turning equipment, table saws, planers, joiners, lathes etc, I'm 100% convinced owning at least a cheaper off shore magnetic base and .001" reading dial indicator isn't in my opinion really optional. There's absolutely far too many benefits to have one over the relatively minor cost once you do understand what there capable of checking.
Run out checks are just part of what they can do. As another example, even router tables with specialized matching router bits for doing complex interlocking joints and once the correct vertical height settings are established with test cuts for the first time, anyone who has a mag base and indicator can then use that, and then with known numbers to write down that exact router bit elevation, and then re- use the same numbers any time the same router bits are used every single time. There's then no more trial cuts needed each time a router bit set like that is set up. There's a great deal of methods used in metal machining that can be used equally well while wood working and for quicker and much more accurate results. Wood working simply has zero need for even .001" accuracy, but the same is not true for the machines and there alignments any wood worker might be using. There's also further checks on wood lathe head and tail stocks I haven't mentioned, and having a magnetic base and indicator would again be helpful. If you happen to own a drill press, tramming and adjusting the table so it's square in X,Y to the spindle rotation is another excellent use. In fact my very first power tool over 40 years ago was a drill press, and the reason I did buy my first magnetic base and dial indicator that I'm still using often. However, even the much cheaper indicators still operate with some very close tolerances, they don't like contamination inside them at all. Their best kept stored away and only used when there isn't a lot of dust present. And don't ever oil the internals unless you know exactly what your doing. Even half a drop of the correct oil is getting close to too much.
