Why can you not use email as it was intended to be used, one address per user?

You could create folders for the two users, then each moves their pertinent messages to their own folder.

Is there anything obvious about your messages that would allow them to be automatically filtered?

Or just advise your users not to delete non-pertinent messages.

We have many emails that arrive from many different companies that are pertinent to different users. Having to separate emails would not work effectively. We could easily do this with Live mail, I am sure it can be done with Thunderbird.

If you use IMAP, then I'd suggest each department is given its own folder. They then move the messages they want to work on into their own folder. Hopefully someone will raise an alert if messages are left in the Inbox, unanswered.

If you use POP then you need to set it to leave messages on the server, and not to delete the server copy when the user deletes. So if department A sees a message which belongs to department B, A deletes their own copy, leaving it on the server for B to deal with. Periodically, someone with supervisory authority would need to review what is on the server and delete messages that have been worked on. I have no idea how you track what has been dealt with.

I suspect you have used POP in the past, but left to itself, Thunderbird will choose IMAP for you. There's no reason why you can't set up the accounts again but using POP.