I'm 26 and don't have (or plan to have) any children. I can't speak for most people, probably, because I personally didn't play an M-rated game by myself until RE4 came out in 2005; but I was pretty squeamish as a kid, so it took me a while to adapt to higher levels of violence and gore in video games and movies. That said, I do know that lots of kids get their grubby paws on M-rated games quite early. I have parents in the classes I teach who openly admit to buying their young children (often boys, though this is a pretty rural, "traditional" area of the U.S.) such violent games, such as Call of Duty and Halo. In fairness, they also buy them games like Mario and Pokemon, so it's not entirely one-sided.
Personally, I don't think most kids under the age of 14 should have access to most M-rated games, if only because their minds are often trying to approximate what they think of as "adult" activities while lacking the generally adult ability of recognizing the game's thematic principles or how the game's representations sharply differ from real adult behavior and decision-making. This lack of perspicacity may lead them to the swear fests, where they think they're doing "adult" things (swearing, engaging in power plays, beating out others) without really knowing what they're saying or doing (and thus being assholes in the process).
At the same time, my fiancee was playing some M-rated games, like Silent Hill, when she was in her teens, and she had the maturity to deal with it (granted, she's always been a single-player-oriented gamer). So ultimately, like so many things in life, it boils down to the context: who is the kid, what is her/his maturity, and how do the parents feel about it? If your child can't help but scream obscenities in online matches at the age of 12, they're probably not prepared for those experiences. Likewise, as a poster said earlier, killing hundreds of people in GTA at the age of 10 may not be the best use of the kid's developmental time. But at the same time, wouldn't they likely find another outlet for their apparent aggression? For some of those kids, these games are an outlet for their adolescent turbulence; but that doesn't necessarily mean we should indulge them entirely because they need to learn to deal with their feelings and not simply vent them.