Actually, a lot of information is made involuntarily public via both gossip and businesses and governments posting or selling information about people.
Normal, though, is a highly overrated and very vague, almost meaningless word. Everyone can define their own standards. The impression of oddness appears when people assume and then get surprised.
Too private to share? I’ve had that impression, such as people telling me about their crimes, intimate relations with others, and especially gossiping about other people. Or just imposing their personal issues without asking first if they can talk to me about such things. Certainly I’ve also been asked for information that was surprising, like people I barely am acquainted with asking me about my personal relationships. Or one person who asked me how I manage to masturbate while traveling with others, and why didn’t I hurry up the life and death process by reproducing and killing people? One of those “fearless (pseudo)intellectual” types, I gathered. Sometimes it even seems to be a tactic to violate other people’s comfort zones, in order to put them on the defensive or make them feel obliged or more intimate than they might otherwise choose to be, which also sometimes goes along with the topic of trying to make the other person feel unfriendly or cold or weird for not wanting to have an intimate conversation.
I don’t think there is any actual subject that by definition is off limits without social agreements, which are rarer than they used to be, particularly in modern and cross-cultural societies. However that just means people are left to define their own boundaries, and to uphold them, which unfortunately isn’t generally taught very effectively.