My best guess is that it uses WebRTC for P2P transfer. It may also use WebSockets to connect to a private room (which only one source-client and one destination-client may join) and streams it.

EDIT: It would appear that it is the latter, in that it uses Socket.IO to stream it to the server and down to the recipient. The server could copy it as it is streamed, and I didn't really delve into the source of the webpage to know if it has any encryption or anything.