It really is a shame that there still isn't a easy way (A person whose computer knowledge extends to using facebook), that I know of, of sending arbitrarily large files that isn't tethered to a specific cloud service and is also reliable (can tolerate connection dropping). It seems that bitorrent protocols are pretty close, but I don't think there is a seamless client that allows for "magical" point to point transact…
Or run a local SFTP server with UPnP and avoid the extra complexity.
In terms of finding each other, the client could get its external IP, open up a UPnP port, and provide the user with a QR code or brief snippet to paste into a chat conversation, which the other user would feed to their client to initiate the file transfer.
You could do this commercially by providing an external website with which the clients could communicate out of band to set up their transfers (for free), and (for pay) optionally provide a cloud sync service (which would help person A "pre-send" a big file transfer until person B was available to receive it).
While some of this is available today, you're right, I don't think there are accessible clients. ICQ, AIM and IRC used to be the solution, as they all did P2P file transfer. But now everything is on the web, so everything sucks.