Your incoming mail will download from the server to the first computer that logs on. To prevent this go into your OE Accounts, select the Gmail account and then Properties. On the advanced tab select "Leave a copy of messages on server"
This will allow both machines to download a copy of the messages however over time your mailserver space will gradually fill up and you will have to manually delete the messages. You could set one computer as above but make sure you check for e-mails on the other first so it gets a copy before the mail is downloaded and deleted from the server.