It slows down for someone looking from the outside in, not for the person at that speed. Hence the "relative" part of "relativity!"
Exactly. Each watch thinks it is the "slow watch" and that the other is the one moving really fast. Hence the problem appears opposite depending on which watch is used as the reference. But both are equally right. Watch A says watch B went :55 in his 1:00 and Watch B says Watch A went :55 in his 1:00.
That's also why noone except cosmologists need to care, because we are always the "slow watch."
If you want to really confuse things you can think of length contraction (objects going very fast appear shorter) in this paradox. A 33 ft T-6A going near the speed of light can appear to fit in a 20 ft garage. But inside the T-6 the pilot would say the garage appears to be only 12 ft long and the T-6 is normal size. Both are correct.
BTW, excuse my nerd-gasm thread jack here. I busted out my textbook, I'm really bored, and I need this...