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 (#990812: Add a "permissions" subcommand to fix/set all file permissions) 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.