The password protection functionality in WordPress is very basic. In fact it's probably one of the weakest parts of WordPress' built-in functionality. All that happens when you enter a password is a cookie is set in the user's browser containing the password they entered. This cookie is not specific to a particular post, it is site-wide.
Therefore it's far beyond the scope of this plugin to determine whether the password in the cookie is correct or not. For example, after a user has entered a password and they are browsing your home page, how should the plugin attempt to determine whether the password is correct for any post on your site? If two posts on the site have two different passwords, the entered password could be correct for one and incorrect for another. Should the log out link show up?
This plugin just provides a log out link for convenience. You may want to look in to one of the many content protection plugins that are available in the Plugin Directory.