Show message to logged user if it have new unread message

If you need to check if a user has unread messages, you can add an Execute custom PHP code condition to your rule:

$account = user_load([site:current-user:uid]);
$unread_total = privatemsg_unread_count($account);
if ($unread_total > 0) {
  return TRUE;
}
else {
  return FALSE;
}