Wow, I've seen quite a few arguments on both sides that seem very skewed or extreme. I think there are some things we should keep in mind in regards to this topic:
1. There is evidence supporting both sides of the argument. Some studies have suggested a positive correlation between video game-playing and violence; others showed no correlation.
2. Yes, it's wrong to assume that video games make people violent because a few crazy gamers became violent. But it's also wrong to assume that video games do not lead to violence based on a few peaceful people who play violent video games. It's the same kind of generalization.
That having been said, my own opinion on the matter is uncertain, but I'm leaning slightly to the side of some sort of correlation. No, I'm not saying that violent video games will make children fantasize about killing people. I don't even quite agree with the empathy issue Trainer Michael brought up. My opinion here centres around the issue of early interest. A person's childhood experiences can have an effect on their behaviours as an adult, though the effect is often indirect. Chances are the person won't precisely mimic the events of such experiences. Rather, it's like a seed being planted in the child's mind. One of several that interact in very complex ways to develop one's personality later on. It's basic psychology.
In the case of violence in video games, it might influence a somewhat carefree attitude toward aggression later on, but other factors--morals the person has adopted, other things they liked to do as a child, other interests, etc--could obscure that. Another factor that could obscure this influence is the video game itself, specifically the attitude it takes toward violence and how and why it's used. If a game lets people shoot innocents without punishment, for instance, the influence could be stronger. Same goes for a game where you play as a criminal, and the criminal acts are emphasized much more heavily than the protagonist's heroic efforts, if any (I'm looking at you, Grand Theft Auto). On the other hand, more light-hearted games where the violence is comical (the Super Mario games, for instance) have the violence be less believable and (sometimes) less emphasized--these games would have less of an influence. Especially if the violence is used only against the bad guys--that is, those who attack you.
But like I said before, these influences will not always lead to a violent person. It depends on the rest of their personality as well, as well as moral influences--which in this case would usually condemn the use of violence. Still, they can have an influence.
This next part is slightly off-topic, but still related. The same concept above can apply to other things present in many mature-rated games, such as sexual content and swearing. Unlike the violence example, it's not as widely understood and communicated that such things are bad. People who hear a lot of swearing early in life are quite likely to adopt swearing into their own language later on, for example.
It can even apply to more positive or not-necessarily bad things. A person who plays a lot of the more cute and comical video games, and/or watches a lot of cartoons, might adopt a more whimsical nature later on in life. (I'm certainly guilty of this. My interest in swords was influenced by video games long ago as well.)
Overall, I believe violence in video games can lead to violent attitudes in people, but mostly if the people were exposed to them as children, and also if there were other negative influences on the person's life as well. And also that people who are into the really violent video games could easily weaken their compatibility with people who don't like them at all. The latter is just a little something to beware of for those CoD fans out there...