Browser to browser communication without a server?

Yes you can try http://httprelay.io with the AJAX calls. It is simple as that:

* Send data: POST https://demo.httprelay.io/link/your_secret_channel_id
* Receive data GET https://demo.httprelay.io/link/your_secret_channel_id

Looks like there may be hope on the horizon: http://www.w3.org/TR/webrtc/