I strip the paint from the handles using a chemical paint stripper, then use sand paper to get any spots I missed. I just cover the rest of the bell so the stripper doesn't get on it.
For snatches, I definitely recommend it. I've gotten heat blisters from bells that had the paint. The other thing is that I do long sets. So, I have to make sure that chalk sticks to the handle. Chalk typically will not stick to a painted bell. So, yes..I would take the paint off the handle.