How can I logout of the desktop app?

You don't log out. You can unlink your computer from Dropbox but this will stop all synchronization (though it will not prevent access to the files you have saved locally). Instead, you should be securing your computer user account. Create multiple accounts on your computer and secure yours with a password so that only you have access to the account and the files it contains.

See: How do I unlink or relink a computer, phone, or tablet from my account?