We already have dozens of apps that do that one way or another.

I wish we had a proper standard protocol for doing this by now. Something maybe based on mDNS, avahi, webdav, and a little glue, so we can implement and run this anywhere. And without requiring internet access at all.

I've actually been having this problem. How can I be assured that my files won't travel outside my wifi, though?

Pushbullet wouldn't want to have to handle all that data, and you would notice very quickly based on the speed of the transfer (unless you had really good upload).

The only way to know for sure of course is to test it. You could try to send a file between two devices on different LANs and it should fail, or you could try it on an offline LAN.