That was different! That problem was caused by Polylang, that plugin had some kind of conflict with Download monitor. I simply find another plugin for a multilingual site.
Found solution!
Problem is that WP automatically block search of protected posts.
I found a solution. It is here: https://clicknathan.com/web-design/include-password-protected-pages-wordpress-search/
Thank you for your answer. I just changed to XAMPP and did not have any problems anymore.
