Outlook does not allow two separate processes to open the same Outlook data store, so you can't just share the .pst file (unless you know that both computers won't be using it simultaneously).
The most elegant solution for email sharing is to use IMAP rather than POP3 to connect to your email provider. You'd need to delete and recreate your accounts, selecting IMAP instead of POP3. This will only work if your provider allows IMAP access.
IMAP basically keeps a master copy of all emails on the server, and keeps track of which emails have been downloaded to which computer, etc. Excellent for multi-computer emailing.