Private profiles allow you to manage who can see your posts. You do not need a private profile in order to manage who can message you, as all messages from users you do not follow appear as message requests.
Conversely, private profiles that you do not follow will not display any information except for the name, biography, avatar and follower/following counts. It is not possible to view the followers/following of a private account you do not follow.
When requesting to follow a private profile, the user will be notified of your follow request.
If the user accepts your follow request, you will be notified in your activity. If they ignore your request, you will not receive any notification.
To switch to a private profile:
To switch back to a public profile: