Question about Thunderbird and Gmail

Hello everyone. I have a friend who has a few PC's, one is hers, one is her husbands, and another is at a vacation home. They have a Gmail account they share and all of the PC's have the latest version of Thunderbird with the Gmail account setup in it. What they're finding is that if one Thunderbird client downloads a message then it's unavailable for the other clients to download. We've checked the settings in Thunderbird and they're all set to leave the items in the Gmail inbox until they are deleted manually. Any thoughts on why this happens? Is this normal and it's just not possible for a shared mailbox to work in the way they'd like it to work? If this just can't work is there an e-mail option out there that will work? Yahoo, Hotmail, etc.?

Sometimes if the first person to read the mail deletes it or moves it to another folder can cause issues like this.

A simple test would be for each user to receive e-mails, and then check to see if they are still on the server. If not, then the setting is not in effect for that user.

I'll have her compare what's on the web mail interface with what's on each PC. Thanks.