, IMAP and Courier

To synchronise mailboxes between different computers and a mail server, you use IMAP rather than POP. Courier runs on the server and transfers messages between your mailboxes there and your local computer.
In's Preferences > Accounts > Special Mailboxes, tick 'Store sent messages on the server', and do the same for Deleted messages. Under Advanced, tick 'Automatically synchronize changed mailboxes' and put INBOX as the IMAP path prefix.
Then on the server you'll end up with ~/Maildir (or wherever your mailboxes normally are), containing .Sent Messages and .Deleted Messages (and a .folder for any new folders you create).
(This works at least with a standard Debian testing set-up).

This isn't perfect though, as Thunderbird seems to use .Sent and .Trash folders for the same purpose.
It seems that, if you create the .Sent and .Trash folders on the server automatically with Thunderbird, then in use Mailbox > Use This Mailbox For > Sent (and the same for Trash), it will use those folders for synchronising from then on (instead of .Sent Messages and .Deleted Messages). In this case, you shouldn't have anything as the IMAP path prefix?