Any ideas? 
If this has been covered before, I apologize, but I searched for “peeing” and “urination” on the site beforehand and got nothin’.
Pavlovian response. You’ve come to associate running, splashing water with urination.
Funny, when some guys hear another guy actually pissing*, they suddenly can’t piss.
I don’t know about women
*Men don’t pee. Men piss. Women pee.
Peace,
mangeorge
This was actually covered in The Return of the Straight Dope, pps. 27-28. Cecil essentially has no solid answer, but points out the stimuli can be auditory or visual as well as tactile (putting the sleeping person’s hand in a pan of warm water), and admits he’s experienced it. His best guess as to why it happens is “the power of suggestion.”