I am sharing a computer with a co-worker. We have two different Trello accounts and we're using the same browser. I need to log out of Trello so I can log into my own account.
Another option (that doesn't involve installing another browser) might be to have one of you use your browser's regular session, and the other open a "Private Browsing" session in a new window to log into Trello. This way you can both stay logged into Trello at the same time (if that matters).