You want to use IMAP! POP3 is not designed for this scenario!
While you can tell POP3 to keep a copy of your messages on the server, it is also downloading them to your local coputer. Note the word "copy". Using this option with a mailbox shared with one person, you'll wind up with three copies of every messaage: one on the server, and one local copy on each of your systems!
Furthermore, it's impossible to share Sent messages in POP3. All sent mail is local.
IMAP, on the other hand, will do exactly what you need. While some data for each message is downloaded locally, the server keeps the original of all messages. Every message either of you access is the same.
In addition, all sent mail is kept on the server.
IMAP will give you a truly shared mail box.