Are you doing freestanding puzzles? I think this is personal preference. I've done it both ways. More often than not, I paint the edges and back.
For the inside pieces, a lot of times I use a #5 blade or larger to cut freestanding puzzles. This will create a larger kerf, which makes the puzzle easier to assemble. It also gives room for paint buildup on the edges. So, with this method, you can paint the inside edges with a thin coat.
When I paint puzzles, I like a diluted acrylic paint, basically the consistency of milk. This stains the puzzle, letting the grain show through. Then I put a spray-on clear coat on it for protection.