If both computers are checking the same email account, whichever
one completes its email download first will make those emails
unavailable to the second PC. To fix that problem, go to Tools,
Accounts, select your mail account, Properties, Advanced,
enable "Leave a copy of messages on server." Depending on how
much server space your ISP allows, you may want to select one or
both of the two associated 'remove' options.
Repeat the above on the second computer.