Nobody likes to ask a naked stranger if he can bum a squirt of toothpaste in order to brush his teeth with his finger. Unless you religiously skip the locker room and go straight home after working out in a neighborhood gym, you'll want to pack a shaving kit with some basic essentials.
To save yourself the trouble of slapping your forehead while you picture the shaving kit left forgotten (again) on the edge of the sink at home, buy a spare that can stay in your gym bag to save the trouble of moving it back and forth.
You won't need anything fancy. For a few bucks, you can purchase a cheap shaving kit bag, a toothbrush, a small tube of toothpaste, a comb, and some deodorant/antiperspirant, plus cheap duplicates of any other hygiene product you use.
This way, you can get off the treadmill, hit the showers, and make yourself presentable before clocking in for work (or going out for a burger). Now, you won't have to ask a naked stranger for anything -- a win/win situation for everybody.