Recently, I started building a website which does pretty much exactly this. When I had it up and running after a few days (the WebRTC API is relatively simple), I found sharefest and have been using that since.

It's a great way to get files from one place to the other. The (encrypted) data does not go via a server, making this potentially the fastest, most scalable and safest type of file transfer available in a browser. One of the extra perks is that sharing files on a local network becomes really really fast. It's miles ahead

Props to the devs for making this! :)