Private Posts/Pages & Search

This is the default WordPress behavior.

http://codex.wordpress.org/Content_Visibility#Private_Content

Private posts are automatically published but not visible to anyone but those with the appropriate permission levels (Editor or Administrator).

WARNING: If your site has multiple editors or administrators, they will be able to see your protected and private posts in the Edit panel.

If you want to show private posts on the front-end of the site to logged-in editors you can add this code to functions.php.

add_action('pre_get_posts','filter_search');
function filter_Search($query){
    if( is_admin() || ! $query->is_main_query() ) return;
    if ($query->is_search) {
        if( current_user_can('edit_private_posts') ) {
            $query->set('post_status',array('private','publish'));
        }
    }
}