Private post and valid user

Hi, I would like to make private post viewable only by certain level user. Although I have done a search on this here, I have not found a solution. I do not mean password protect post. I have read the documentation and for private posts, I understand that those posts will not show up except in the admin area under edit.
I do not want password protect because then that “This post is password protected” text shows up and I do not want that.
Since there is a way to query the database to show post in categories, is there a way to check for user level and display those marked private? Would it require a template?
Thank you in advance for your time.