Actually, if you want, you can even make both use the same IMAP folder (Thunderbird can be configured to use an IMAP folder as its sent / fcc folder).
But to answer your question, you'll need to be using IMAP on your PC. Then, you should see (or at least be able to subscribe to) the "Sent" folder over IMAP.
I don't know the exact commands to do this in Thunderbird, but basically, then you'll want to tag all the messages in the IMAP folder, and drag them to your local sent folder.
Hope this helps; maybe someone who uses Thunderbird can give more specific instructions (the kbase instructions on setting up Mozilla should be pretty close).