Today, I am experimenting with a cool project called offline browser communication. You can check it out here. It requires one side to be a native client, but the demo allows for connecting two people on the same network by agreeing on their mDNS Names and DTLS certificate beforehand.
This feature could easily be integrated into web browsers, but it would need support from spec bodies since WebRTC currently does not allow users to select their mDNS Names. However, using the host's mDNS name could still make it work smoothly :)
An interesting twist is that mDNS can also be used to identify hosts' names, which opens up possibilities for combining both functionalities. If you're curious, I have a jsfiddle demo available here.