Sending system e-mails do not work in spite system says it sends e-mail So, my Drupal 7 does not dispatch those messages (notify user on registration and such) in spite the system says it sended e-mail. Other scripts (like send the web page by email) work well, though! 

I am on dedicated unmanaged (i manage it ;) ) Debian server with LAMP, and for e-mail I installed Postfix.

If someone got clues i would be grateful!
If the system log reports it is sent, mostly likely it was sent. Perhaps those mail went to SPAM box? If that is the case, check if you have Reverse DNS set up for your domain. 

Besides, you can utilize 3rd party email service (such as Gmail) as your Drupal mail system. 
Check out SMTP module.