How to – one email address checked by two users

POP is working as designed, here. If you access a mailbox via the Post Office Protocol, you are expected to be transferring it off the server to storage of your own. (RFCs 1081 and 1939 both make this point explicit in their introductions.) From the server's point of view a message that has been RETRieved but not yet DELEted is indeed "read" by the POP client.

If you are using the Post Office Protocol to keep mail on the server indefinitely, then you are in fact abusing the protocol. POP has never required that maildrops hold mail indefinitely. Indeed, RFC 1939 § 8 is clear about the fact that servers are quite free to outright delete messages that you've retrieved but not deleted yourself, let alone simply mark them as read. The Post Office Protocol is, as the name suggests, a protocol where mail is held at a "post office" drop point, from which you collect it and take it away whenever your MUA is connected to the network.

So your question is essentially unanswerable. You are asking how to make POP not work as designed. Of course, this is another case of picking the unsuitable tool for job and then asking how to modify the tool so that it works. If you want a protocol that's designed around the paradigm of having a remote mailbox maintained on a mail server (rather than POP's last-leg-of-the-store-and-forward-chain paradigm) which is accessed by multiple client MUAs, then use IMAP. It's that simple.

You are right, IMAP tracks opened emails centrally, along with reply and forward flags. As there is only one email, there is only one place to store these flags.

One option would be to instead of having a shared email, forward any incoming emails to the address in question to all members that need to see it. That way they can have their own flags set for the emails. You can use rules to ensure they are kept separate from other emails. Of course, this means that any outgoing emails would be stored in the users own sent items rather than the central email sent items (though you can often work around this as well).

An alternative would be to use a different indicator for unread than the unread indicator. For example, most email clients have a method of marking an email in some way. Outlook uses "Follow up" flags, Thunderbird uses tags - important, personal etc.

You could have any emails to the central account be flagged upon receipt within each email client with a Follow up flag using a rule. When a person reads an email, they remove the flag (usually this is a single keypress - and in some clients would be automate-able with anther rule). That way they have a clear indication of which emails they have read and which ones they have not independently of the IMAP unread status.