This should work with WebRTC (http://www.webrtc.org/) using the reference app: https://apprtc.appspot.com/

It should enable you to connect 2 browsers (e.g. Chrome and Firefox), all you need is to transmit the ID of the chat ("r" parameter from the chat URL you will get) to the peer, or the full URL.

It keeps amazing me that we do not have simple JS solutions for this, when projects like Chatroulette can be built in "2 days 2 nights" according to the author.

(by the way, here is WebRTC support for other browsers: https://code.google.com/p/webrtc4all/)