Thinking about it (which hurts too much this close to new year ) I think this is actually the opposite of the USB scenario, you have two devices that are both able to connect to other things (websites) but cannot be connected to, this makes them the same as a USB host, so all you need to fix the problem is a USB LinQ

But seriously there should be a pretty easy fix that should be small enough for you to host almost anywhere, assuming:
- The upload mechanism the same as the one used here for attachments works OK for both of you
- The download mechanism that most places use where you click a link, it displays a page showing that a download should automatically start, then the download starts works for both of you.

What then happens is the file 'receiver' goes to the a webpage, and a download is triggered 'automatically' at the same time (well give or take whatever the download timeout in the browsers are, I am guessing you may have a couple of minutes) the 'sender' chooses the file and uploads it from another webpage, each uploaded chunk of data gets immediatley sent to the person dowloading so there is no data actually stored on the server. Actually that does sound a lot like the USB solution, something dumb in the middle just forwarding data.

I guess it should be a fairly simple task for someone who knows that sort of stuff, or has acess to the upload and download examples I gave, I may be interested but I am hacking USB at the moment