What options does a web developer currently have for implementing client-side peer-to-peer systems in the web browser? Some hypothetical examples might be a file-sharing service that bypasses the web-server after clients find eachother, or a chat service.
the site chatroulette uses some flash technology as discussed here.
And I am also aware of Opera Unity which extends the browser model to encompass both a client and a server, as well as this firefox plugin which works similarly
I also just came across this blog post related to the Joose js framework, but am not yet sure if it is relevant here.
Support for this kind of thing seems like it might or might not be supported by websockets in HTML5 and has been discussed here and here and here.
I recommend you to take a look at RTMFP protocol in Adobe Flash Player - it seems you can build something close to what you're asking with it.