Your privacy is not a thing in and of itself, that you either have or don't. Privacy is your ability, and capacity, to choose for yourself what you share about yourself, and when, and with who. And the other word we have for that is "dignity".
Privacy is the agency you have over your dignity.
"A private social network" only makes sense in through this lens, of people wanting a social environment where they have agency over what they share, when, and with who.
It's a reasonable thing to want.