I'm beyond it now... but when I used a Diva cup (and before that a Keeper) my hands never got messy.

The two things that were somewhat of an issue in public restrooms were (1) remembering to wash my hands BEFORE going into the stall, and (2) wanting to rinse the cup before re-inserting it. Without rinsing, it isn't perfectly clean on the outside after emptying, so whatever I was wearing would get a little stained unless I wore a liner (which kind of defeats the purpose of the cup). Staining was always really minimal and not at all a concern with cycling shorts. Also, rinsing before insertion helps lubricate it, clean or not. I guess you could bring a water bottle into the stall to squirt it off, but I never tried that.