Right hand.
When out in public, I always wash my hands after peeing. I usually do at home too although there could be an exception if I have just had a shower and there is no way that anything I touched is remotely dirty. I might not bother.
I virtually never get actual pee on my hands. That is not a factor.
As a rule, I feel that society is probably a little over worried about the post-pee handwash. Like, if you look at the actual logistics of it. I think there is an exaggerated idea of how dirty Johnsons are. It's probably a bit of residual Puritan thinking. But I go along with it.
I am very proactive about cleaning up stray pee. I usually pee sitting down so it doesn't happen often but if there is any possibility, I take care of business. I'll wipe the rim and whatnot without blinking an eye. Then wash my hands. Can't recall ever misfiring to the point where I got some on the floor. But I would go there if I had to.
You better fucking believe it.
