I found that it was the fault of my template. I am using one based on Blue Mist 1.5 by AOE media GmbH. In the comments.php file I removed the part where it checks the post password with the cookie value.
It starts with:
if (!empty($post->post_password)) { // if there's a password
I replaced it with the default:
if ( post_password_required() ) { ?>
<p class="nocomments">This post is password protected. Enter the password to view comments.</p>
<?php
return;
}