I would guess that the machine stealing the messages has been set up to use POP. My usual advice to someone wanting to work an email account through multiple computers would be to consider using IMAP. IMAP stores messages on the server, so each connected email program sees the same sets of messages and folders. Some folk dislike IMAP precisely because it stores your messages on a server, preferring POP because it grabs the messages and stores them on the local computer. That works fine if you only use one computer. You could set the machine configured to use POP to leave copies on the server. You might then get into a tangle trying to keep track of which messages you have (or don't have) on each computer.