|Map shared drives on each of the computers. Then copy from one shared drive to another.|
That is assuming that you are talking about Windows machines. If they are Linux or UNIX machines then you could SSH into one and then FTP the file to another computer, or copy it using NFS or SMB shares. You could even do this with Windows computers if one of them was running a telnet or SSH server. There are other possibilities, but these are the most common ones.