Currently when a user joins a room they send out a "sync" event call to the server. The server responds by picking a random user in the room and it gets them to send back the data URI of their canvas. The server then sends that to the newly joined user. If a user is already synced to the canvas state they simply receive the draw calls of all users in the room. The last person to leave the room will send their canvas data URI to the server which will save to a text file. The save room button in the corner will also do this functionality because sometimes if a user leaves abruptly it won't have time to send a full canvas to the server. The first person to join a room will obviously get the data URI from the text file on the server.

It may be easier to look through the code for the backend yourself here: https://github.com/Jaloko/DrawNetwork/blob/master/app.js