It uses a P2P connection when WebRTC is supported by the browser. (WebRTC needs a Signaling Server, but it is only used to establish a connection and is not involved in the file transfer).
If WebRTC isn’t supported (Safari, IE) it uses a Web Sockets fallback for the file transfer. The server connects the clients with a stream.
No files are saved on a third-party-server.
No. ShareDrop is built with ember. Snapdrop is built with Polymer. I wanted to play around with Progressive Web Apps and then I got the idea to clone Apple’s Airdrop. By doing research on this idea I found and analysed ShareDrop.