List Category Posts plugin uses get_posts to actually get the posts and its default post_status is publish and that is way you won’t get any private posts.
To “Fix” it you can edit the file named include/CatList.php of the plugin and add
$lcp_query .= '&post_status=private';
before line 51 before
$this->lcp_categories_posts = get_posts($lcp_query);
this will get you only private posts and to really “FIX” it you need to add this code:
if(is_user_logged_in()){
parse_str( $lcp_query, $lcp_query );
$lcp_query['post_status'] = array('publish','private');
}
on line 51 again before
$this->lcp_categories_posts = get_posts($lcp_query);
and offer the author of the plugin this as a patch so the next time he updates you want lose this fix.