My understanding of http or https browser sessions is that they are terminated upon closing of the browser by the incoming user (i.e., you when you close your browser).
Question remains, can we transfer files over this connection? I'm guessing, "no". Teamviewer TechSupport should be able to provide us with an answer on this. Can we submit questions to them from here?
In a few days I will do some more testing of the following solution:
Transfer FROM local TO the remote machine: Post the desired file to my web site. In a subsequent, remote TV session, open a browser session on the remote machine, and pointing to URL where I posted it.
Transfer TO local FROM the remote machine: Same procedure, in reverse. Probably install Dreamweaver (DW) on the remote machine. Use DW's implementation of FTP to perform the upload, again to a known location on the Web. Download to local machine using DW or web browser.