Zotero downloads all your data to the computer you use, so there is no such thing as "logging off." (Just as you can't log off from your Dropbox sync folder or Thunderbird e-mail, for example). If you want your data private, you'll need to use a separate user account on the work computer (you don't have that?) or, if all else fails, you can create a password protected profile on Firefox--https://support.mozilla.org/en-US/questions/807379 (though note the explanation there about how basic that protection is).