Hi, first of all as infa, because I don't know if you are aware, only the user himself and the administrators can see this information from other users. It is therefore always worthwhile to create test accounts with the various memberships. But to answer your exact question, yes that is possible.

u go in ur Studio under the app: " Develloper" - "Fields" - "Persons" - "Edit Persons"

Screen 1

here u see a ll fields and can manage this - if u want that users can set the privacy, than open e.eg Birthday and set the Privacy settings like my 

Screen 2

it will looks like my Screen 3

Screen 3

But when u make this setting, than u should do it for Add and Edit Profile