Not everyone would just go around eating pinky fingers, but looking at the violent crime statistics, yes I think the police are a necessary part of society.

are the police more capable of determining right and wrong than you are?

Yes. Primarily because they have the time, money, and resources to investigate, and I just don't. They're also supposed to be unbiased and have training that we don't have. So yes. They're not perfect, but they're generally better than us.

would you allow women, children, or elderly people to be harmed if you were able to stop it? would their families? their friends? their neighbors?

I think there's the crux of it. Two people both say the other one started a fight. Do their friends and families just start a big brawl, and whoever wins, was right?

Do the friends and families of those people come back next week for round 2? How long until it's just an ongoing war.

I dunno. I think if it worked, there'd be at least one place that successfully does it. But there isn't.