If we save 'file_private_path' variable on 'admin/config/media/file-system' page as empty string, private file system becomes unavailable in Drupal: http://drupalcode.org/project/drupal.git/blob/b9127101ffeca819e74a03fa9f...
The same should be done in Drush.
Initial patch for branch 7.x-5.x attached.
Why I decided to fix it:
drush perms command ( ) takes private path '' (if it is an empty string), convert to absolute path and sets permissions for whole drupal directory as for private folder. Not so cool.