I don't believe that any kind of sharing, in the way that you want to use it, is possible with the Mozilla (Netscape, Firefox, Thunderbird, and derivatives and variations thereof) products. It is my understanding that the foundational architecture of these programs was not designed for simultaneous multi-user execution. One may
be able to cobble together a 'network' wherein there can be sequential (that is, non-concurrent) access, I would opine that this is more of an academic exercise than a practical one.
Now for the good news. "Is there a way to use Messenger on both computers at the same time?" Yes - with the proviso that you are not actually looking at the same mail folders.
Forget the sharing - that is, don't try to share at the Netscape level. Below is the technique I use. But first, will this serve your purposes?
On one (or both) of your computers, in Server Settings check "Leave messages on server". Also consider checking "Delete messages on server when they are deleted or moved from Inbox". The users on both computers will have to have an understanding as to who is to dispense with which messages. But I think that this may satisfy your underlying requirement. And yes, both computers can
use the same POP3 account.
Now, concerning sharing. In my home network all computers have the Mail folders fully XP shared (this could be an exposure if we didn't trust each other, and if we weren't behind a firewall from the internet; further constraints can be applied in XP Pro, not in Home). If I need to transfer a message, intact, from one computer to another, without having to Forward it - and thus pick up more confusing headers, etc. -- I Move (or Copy) it into a mail folder I've named "Xfr". On the other computer, with Thunderbird (I used to do this in Netscape, too) closed, I do a file Copy of Xfr. I also delete any pre-existing 'Xfr.msf' file on the 'incoming' computer. When the mail program is opened, the Xfr mail folder is available for processing.
This technique works between Netscape and Thunderbird too, since the actual message format within the mail folder is the same; I would guess that this is common across all Mozilla-based mail products. I also used the technique several years ago when transitioning from Netscape 4.8 to 7.x, but the transfer medium was diskette - before the network was installed.
UserAgent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:188.8.131.52) Gecko/20061206 Firefox/184.108.40.206