To logout, click your profile icon at the top right and select "Log Out"
Note: this will log out of all Trello sessions in the browser, so you and your coworker can't be logged in at the same time unless you are using 2 browsers or are browsing incognito.
Take a look here: https://accounts.google.com/b/0/IssuedAuthSubTokens
and if you login to gmail, you can see your (mail) account activity + detail-link in the bottom right corner ('detail' is a link)
Last account activity: 19 minutes ago
The "Device activity & security events" section of https://MyAccount.google.com will show you all of the devices that have been used to log into your account recently, but you can't force a logout from there. The only way to logout of other devices is to change your password, and that will log you out of everything except the device you use to change ...