Well, you can with FTP and UPnP. You the client gets the external IP of the machine, then use UPnP to open ports on the router/firewall, then uses PASV and connects to the destination. The destination runs an FTP server, that uses UPnP to open up the incoming port, and gets its external IP and uses it for PORT mode. The end result is two individuals behind NAT sharing arbitrary files in bulk. You can even tie this into FXP to do syncing with the cloud.

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.