WordPress – allow password protected content in search results
Source: https://wordpress.org/support/topic/how-to-allow-password-protected-content-in-search-results
SOLUTION:
Add this code to your functions.php file and you will get it to work.
add_filter( 'posts_search', 'include_password_posts_in_search' );
function include_password_posts_in_search( $search ) {
global $wpdb;
$pattern = " AND ({$wpdb->prefix}posts.post_password = '')";
$search = str_replace( $pattern, '', $search );
return $search;
}