I'm a girl about twenty, and I work at a GameStop with only guys for coworkers. I have to say, while they're a lot more competitive than I am, it depends. Most of the guys there really like co-op, and another guy prefers multiplayer to shoot off their heads. Still, all in all, they're in it to win, while I'm perfectly content to be a team player so long as I have fun. This is, however, in FPS land, where I acknowledge I'm pretty average, and so when it comes to things like RPGs, Rock Band/Guitar Hero, MMORPGs, I become a lot more competitive. So while I think men are in general more competitive, on top of that, I think the spirit of competition can become a lot more relative.
Sometimes, though, I WILL get called out for being a girl gamer at work by some of the more asshole customers who think I don't play, but I'm just a token of the store of some sort. Online, it's difficult to avoid getting hit on (especially on my troll, strangely enough), and sometimes on mic, they'll think I'm a pre-pubescent boy before it dawns upon them that maybe I have lady parts. Whether or not they question my skill, well, usually the rule is you don't piss off the tank.
Rather, I'm more interested in whether or not anyone notices that girls are often thought of as the cloth-wearing mage or healer. Personally, I feel naked without plate armor, and form loving bonds with my axes. Is it that girls in general really do prefer to take classes that get their hands dirtied a little less than a melee fanatic, or is it a stereotype given to us 'cause casters wear dresses? >.>
I swear, one more mother who buys their daughter My Little Pony: Pinkie Pie's Party because The New Super Mario Bros. is too "boyish" and I'm gonna vomit rainbows. "Girl games" tend to be an excuse to sell mediocre/crap games painted pink.