By now, Penny Arcade's comic has become mantra on the internet: a normal person, given anonymity and an audience, will become a total fuckwad.
It's actually a lot more complicated than that formula, though I suspect the simplicity of the idea is what made it catch on in the first place.
An article on Motherboard delves into the complexities that come with our social networks, and how they do or do not influence our behaviour. They found that studies show little correlation between destructive behaviour and anonymity.
Alex Haslam, professor of psychology at the University of Queensland, had the following to say on the matter:
There is no evidence of a relationship between anonymity and abuse. Everything hinges on moderators, and the key moderator is, what is the norm of the group? If the norm of the group is to be destructive, well actually then anonymity can enhance that. But if the norm of the group is to be constructive, then anonymity can enhance that. So it isn't the case that anonymity is one of those things that feeds into brutality.
He goes on to explain that the expectations and norms of our respective groups dictate how likely we are to act a certain way -- which is to say, anonymity can act as an enhancer for both kindness and cruelty.
It depends on who we associate with and who we think is watching, though having an audience, period, doesn't sway your behaviour in a certain way. You tend to gauge what you think that audience wants, and are likely to act in a way that's beneficial to you. What is beneficial to you is not always evil.
The whole thing is worth a read, especially for those of you who are interested in psychology and human behaviour.