How to set up Drupal behind a proxy correctly?
The problem has to do with how cookies are associated with domains. After you log into a Drupal site, Drupal sends a cookie to the browser that it uses in future requests to show it has been authenticated. Drupal associates that cookie with the domain that the login occurred on. Since your proxy sends requests to your Drupal site using the hiddendomain.be domain, that is the domain that Drupal associates the cookie with. However, the visitor's browser sends requests to publicdomain.be, and so considers the cookie it received from your Drupal site to belong to that domain. It is this discrepancy that is causing the problem. 

To fix this, you need to configure Drupal to associate its cookies with the publicdomain.be domain. In your settings.php add, or update, the following line: $cookie_domain='publicdomain.be';

If your public and private domains share a root, then you can use the domain root as the cookie_domain. For example, if the public domain was www.example.com, and the private domain was private.example.com, then in settings.php you would have $cookie_domain='.example.com'; (note that there is a dot before the domain).