I currently own and operate two specific grinders that do a superb job on reel mowers. I've sharpened hundreds of homeowner and dozens of commercial units. When I first started learning about grinding these old fashioned mowers, I took some valve grinding (or lapping) compound, which is available at many hardware or automotive supply stores, lightly spreading it onto the reel mower's blades and turning the bed knife adjustment screws so they contact and slightly wear into the knife edges. I would then push the unit up and down a sidewalk to obtain this wear in pattern, adding compound or tightening the knife engagements as needed. In this respect, most reel mowers are "self-sharpening" and eventually get to a point where they really should have a relief grind put back on the reel knife edges. That is what I grind now in my shop. Until that time, keep in mind the bed knife must lightly contact the reel, like a pair of scissors. After using the compound, wash the mower off and apply some medium to light oil on the blades, then adjust the bedknife contacting the reel as lightly and evenly as possible.
Some golf courses have a machine that spins the blades backwards (called back-lapping) and they pretty much do the same thing, and eventually they will need a relief grind sometime, too. I've also noted some older reels are of superior quality steel, much better than the current hardware store versions available today. Many of the Scotts mowers have thinner blades but have good steel in them. Watch for loose bearings, wheels and bushings if looking at rummage sales to purchase a good unit.
I've enjoyed seeing really old mowers from the past, incuding several from around the turn of the century. Each machine is similar, but the styling and design of the frames, wheels and mechanisms are varied and interesting.
As another point of interest, I typically charge just under $30 to properly sharpen and grease/lube and adjust a homeowner's reel mower.