Mac Mail -- synchronize IMAP sent boxes
First, check in the Mail.app preferences:


Accounts -> Your account
"Mailbox behaviors" tab
the "Store sent messages on the server" checkbox. If checked, the sent messages are stored in the server, if not, theyre stored locally.


If you storing sent messages on the server, probably: your IMAP-server account uses different mailbox for Sent emails as Mail.app uses by default. Therefore:


you should find the correct mailbox by checking all available mailboxes in the IMAP-server account in the sidebar "Show" (usually some names with "Sent", or "sent-mail" or like).
click on the folder
In the menu: "Mailbox" -> "Use This Mailbox" and select "Sent"
Now, your Mail.app will use this mailbox for the Sent items.


If the above not helps (and probably won't because, as you write, you "can't browse the server mailboxes) you should ASK your IMAP-server admin what IMAP-server "PATH prefix" is used by server. Probably the admin setup some unusual configuration, you probably will need enter something like "/" or "INBOX"...

Warning: Don't experiment with different IMAP PATH settings because you can mess-up you current local mail cache. If you still want experiment (what i strongly don't recommending) you at least should first select "Don't keep copies of any messages", what will delete you local mail cache). When you will have the correct setting you should enable again the local chache and the Mail.app, click Synchornize and the Mail will reload all messages again from server. (hopefully). You've been warned - don't blame me if you loose all your mails. ;)