I need a small double-ended shaft with accurate, coaxial ends and a shoulder in the middle for a couple of ball bearings to rest on, so thought I'd try turning one between centers, which I've never tried before. The beauty of this method is that the shaft can be removed, measured, flipped, or whatever, and can be re-mounted in exactly the same position.
I first turned the shaft blank to length, center-drilled both ends, and turned down both ends to slightly oversize, so less material had to be removed when mounted between centers. The headstock-end center is a small bit of steel bar held in the chuck, turned to a 60° point, with a screw sticking outthe side to engage the drive dog. The drive dog itself is a bit of aluminium bar drilled and split as shown so it can be clamped on the workpiece and engage the drive screw.
The setup worked beautifully, although I found it best to take very light cuts. Once one end was machined, the shaft was flipped and the drive dog mounted on the other end, with a bit of aluminium shim under the setscrew so it didn't mark the finished shaft.
Obviously, the tailstock center must be aligned with the headstock, otherwise you'll have a very slight taper. Mine wasn't great, but it was sufficient for the job.