I am aware that an early draft of HTML5 specified peer-to-peer connections using the PeerToPeerConnection() constructor. However, this was replaced by WebSocket which, to my knowledge, does not support peer-to-peer connections.
In addition, the device element specifies a peer-to-peer connection interface, however no web browsers have implemented it yet.
Opera Unite allows Peer to peer (basically gives your browser web server, file system and nat-traversal capabilities), but this is Opera only.