The POP + SMTP protocol doesn't support SENT ITEMS sync, regardless what server or client software you are using (it's not a SmarterMail issue, every other mail software out there works the same way).
It only DOWNLOADS (fetch...) new mails present in INBOX. It doesn't care about other folders.
NOTE: "download" is not "sync"... And it only download NEW mails...
If you want to sync up with the mail server you MUST use one of these protocols (note that everyone have a slightly different way of syncing... You MUST study and understand how they work and what you want to achieve...):
Some other software may use a proprietary way to sync clients and servers.
For example, Kerio Connect uses a plug in called KOFF to sync the server with MS Outlook on Windows PC, but it needs to be installed and configured manually on every Windows PC that use MS Outlook...
Gabriele Maoret - Head of SysAdmins at SERSIS
Currently manages 6 SmarterMail installations (1 in the cloud for SERSIS which provides services to a few hundred third-party email domains + 5 on-premise for customers who prefer to have their mail server in-house)