Generally, a head-end center and the tail center should line up point to point when brought together, looking down on them and horizontal at them. If the lathe bed was perfect that line-up would continue to the length of the lathe, if not the tool bit would be higher and lower on that part at different lengths parts. With a large part, perhaps 1” in diameter set between centers you should be able to one hand push and pull this part at both ends and see that nothing moves. The saddle should be tight on the V way for the whole the bed, it should not be riding on the top of the V but riding on the angled sides of the V. Often you can Sharpi blue the V way to see if the bed is tight all along the entire distance.
If the lathe is tight in the manner a near straight part should be possible even with great wear on the machine. The method is to measure the part before you do any turning and mark it zero-zero. Or + .1,2, 3 whatever. You can use a rounded nose tool bit for a touching gauge. you set the part between centers and come in with the tool bit to just touch the part ( eyeball, touch a shim,) touch the part looking at you dial you can that both ends of the part match at the place where to real tool bit will touch/cut the part. Yes, you then adjust your tail to make the part where it will be turned straight.
*This simple test should get you better than + .001 without not taking a cut.
A lathe that is an overly worn-out machine may make a varied part diameter along the part but at least the ends of the part will be the same.
*Trying to use an indicator to straighten a part will often fail because the indicator point will/may not touch the part the same as the tool bit will touch the part.
You should have the part/tail adjustment to +- .001 before you touch the pat with a tool bit and everything about the machine and the part should be on hand wiggle tight/solid.
With both ends the same diameter on a wore out machine you may need to abrasive paper the mid area of the part to make it perfect.
Oh, if a live center or a center held in a chuck has a wobble the the wobble end of the part will be smaller.