Is the data private or not?

A question. I have an issue understanding privacy of data, which I have been trying to wrap my head around. Hopefully someone here can help me.

As I understand it I can keep my data completely secret from anyone to see in a blockchain, which means I could if I wanted store my passwords there. (Maybe not recommended.) Unless someone gets a hold of my private key it´s safe with me. If so, is that valid for both private and public blockchains?

Or I can keep my data completely open, which would be a typical private blockchain case if you i.e. want anyone to track the origin of a tuna fish. Is that possible for a public blockchain as well?

I understand there are many different implementations and versions of both private and public blockchains, but I was looking at a more general case.