I never take it off, and I don't care whether the scanner(s) beep or not (neither has anyone at TSA, in my limited experience). I don't recall them beeping (usually have those backscatter scanners anyway, which haven't beeped about a watch), and no one has ever asked me to take it off at security.
I travel once or twice a year internationally, through Korean, Japanese, & Canadian airports, and regional US airports like Detroit, La Guardia, Newark, Chicago O'Hare, Raleigh-Durham... never had a problem. I never ever (truth!) take off my watch or any other jewellery.
(Not trying to brag about travel or appear to be an expert, just trying to give a range of the places where I haven't encountered a problem!)
To be honest, I have more trouble making sure every last drop of water is emptied from my water bottle (a stainless steel double-walled bottle, which goes through the conveyor-belt fed scanners just fine) than with watches/jewelry. If you leave even a drop in there, it seems, TSA can see it, and they'll fuss. hehe