I suppose the same account is set-up on both computers. What's the account type on each computer - POP or IMAP?

Both are IMAP.What now?

Both are configured using IMAP. Now What?

Then what you see is the expected behavior. With IMAP all mail is managed on the server, and what you see in Thunderbird just mirrors the status on the server.
This isn't specific to Thunderbird but inherent in the IMAP protocol.

Thank you, that makes sense, but is there a way of dissociating the two computers without affecting the source being the server. What are my options. Must each PC have their own separate addresses? Please spell it out, and thank you for clarifying this.

Imap wasn't designed for this kind of multi user situation. It assumes that you the account owner will be using it to manage one account via multiple devices. So once you have read a message it has been read, regardless of which device you were using at the time.

You appear to be describing a situation where you want more than one user to read each message, without anyone knowing if anyone else has read it or even dealt with it. Frankly, I can't imagine a set of circumstances where this workflow would make sense or be useful.

You can do various things such as manually setting messages as unread, or moving them from one folder to another to indicate that they have been processed, but much of this relies on users being thorough and conscientious.

You may need to be running your own server to do this sort of email message managing, if you really need to fork a message so it gets to multiple recipients. Managing which users see a particular message is a job for a server, not the end-user's email client.

I should add that if you operate the account using Pop then messages can be left on the server without any indication that they have been read. You set each client to leave them on the server.

However if you don't know which messages have been read, I don't know how you will know which ones to delete and when to do it. Without active maintenance your server could quickly become full.

Thank you very much. Very well explained. Thank you again.