The Google Integration Service does not log into anyone's Gmail account but it's own.
Salespeople (or any internal user, actually) just BCC Act's e-mail account, the service sucks up the message and deposits it in the appropriate contact's history based on the TO address, and the message is identified by the sender's signature in the message. Unfortunately, it does not seem to capture recipient or sender e-mail addresses in the history, but that seems to be just the way that Act! does it. I did not confirm that it also puts the message into the sender's history, assuming that (s)he has a contact record, since that is less important to us, but it might.