Screen sharing: how do I send a file from remote computer?

At work, I use 2 Macs, each of which is connected to the hospital network via ethernet.

Using screen sharing, I have successfully been able to control one Mac remotely using the other Mac. Cool!

What I would like to be able to do is to send a file from my main computer to the second computer which I am remotely controlling with the screen sharing.

I tried the most intuitive thing which was to drag and drop the file to the shared screen window. However, this did not work. I also tried copying the files to the clipboard and then pasting it from the clipboard to be shared computer using the menu commands from the screen sharing window. This didn't work either, resulting in an error message indicating the file is not available.

Is there a way to do what I am describing?
As far as I know my "sharing" settings should be set correctly since I've enabled everything.

(Ultimately, I was able to accomplish this by first uploading the files to MobileMe's iDisk then accessing MobileMe from the screen sharing window. This worked, however, it took quite a while and I am hoping there is a more direct way to do this.)