My experience is that people not only do not care about privacy but look at people who care about privacy as having something to hide or paranoid about something that simply doesn't matter.
Private mode has very little impact on what you share with the world/through the network. It's there to keep your browsing private from people you live with etc., not from anyone else.
I don’t even browse in “private mode.” Not because I don’t care, but because I assume it won’t really change anything.