I made something similar last year: A p2p chat popup (that sits on bottom right of the page as a bubble) that allows all visitors to chat with each other. It had simple keyword based moderation in-built (can be easily bypassed though).
Was planning to add github oauth to get a known identity and persistent messaging so visitors can chat with each other across sites.
Instead of a webmaster adding script to their site, it was a browser extension.
The intent was two folds:
1. Get to know other people having similar interests,
2. Try something on the lines of a decentralized chat/messaging system.
It sounds like a nice idea, but probably only one of those things that work only once at scale - with hundreds of millions of pages, you will only meet others on the most common ones unless the extension is already very popular?