The WordPress private page functionality only allows for administrators and editors to see the private content.
If you’d like to offer a private page to another user type, like an author or subscriber, you have the option to generate a password–protected page. In that situation, any user with the password receives access to the private page.
Another option is to install one of the many free or premium private page or content restriction plugins. They often include features for adding unique user roles to the list of people allowed to view the content.