It depends whether the mail clients have been configured to delete the emails from the mail server as soon as they've downloaded them or not. Most aren't by default...
If the e-mail client running on the first PC to connect is set to inform the server to delete the e-mail once a confirmed download has been performed then only that PC gets the e-mail. If it is set to merely confirm receipt but not delete then the e-mail stays on the server.
Then ditto for the second PC.
If neither deletes you may need to go to the Internet service and delete the e-mail there 'by hand'.