Mark,
A great question! I do not use a reaming machine/bench. This is not to say that it is correct, I really wish that I had one, but the shop space is too small. I am currently reaming an original barrel. This barrel is a nice one, 15/16 straight, 0.38 caliber, 38 long (one of those later percussion barrels that give a perfect definition of "nose heavy". I plan to rifle the bore at about 0.43 caliber and shape the outside to a swamp then put it into a G. Schroyer gun.
1. For the rough boring of either the hand forged bore or one already rifles, but corroded badly. I use a twisted square reamer, 18th c style. I usually drive it with an electric hand drill variable speed on slow, or, if doing a show-n-tell, with a hand brace. Do not use a lathe or other rigid setup as this may force the cutter off bore center. Doing the job holding the tool by hand gives the needed flexibility. If you use a modern twist drill, you will be sorry! The modern twist drill will cause a lot of trouble and very often go drilling off center into a big mess - modern reamers are even worse (don't ask how I know).
2. For final smooth reaming always use an 18th c armory reamer. Usually the first few passes with the reamer I do with a hand brace. Later passes I often use the variable speed electric hand drill (if the crowd is not watching). Photos of the armory reamer can be seen on this topic: http://americanlongrifles.org/forum/index.php?topic=19962.0 (Barrel Freshening - HDTDT)
3. One modern tool that really does work well for this type of bore reaming are the piloted drill bits sold by Midway USA. These tools are specifically designed to ream gun barrels prior to the installation of barrel liners. They are in various sizes from 3/8" - 5/8" diameters. They have a replaceable pilot section with which the tool is guided on bore center and will not wander off center. One drawback is their expense, about $80 per drill bit, but I think they are worth the cost.
Jim