"Sent messages only appear on the PC from which they were sent" Yes, the other pc has no way of knowing what you do on the other pc. Workaround: You could send yourself a copy of the sent email and then collect it on both pcs.(only other way to sync them is to use IMAP instead of POP, so that all folders on pc will sync with the Server including sent emails)
As for downloading/receiving email - on both pcs you need to: OE | Tools | Accounts | highlight the email account, click on Properties | Advanced tab, and tick 'Leave a copy on Server'. So that when you collect email it is available on server for both.
If your received emails are 'disappearing' you need to: OE | View | Current View | and put dot in front of 'Show All Messages'