When Private mode is ON:
Only your name, profile picture, wallpaper, title and location will be visible to players who either:
You have on your Follower list
Have you in their Follower list (if you've accepted them)
Only players that are on your Follower list or have you in their Follower list can send you direct challenges and chat messages.
Your minimal information will be visible in the following places:
If you press "Play Now" you will meet up with other players. They won't be able to send you chat messages.
Rematching often does not work while in private mode.
Note that if you add someone to your friend list they will see your extended profile even if they haven't sent you a friend request.