It depends on whether all private windows have been closed. If you open a new private window when another is already open, you remain logged into sites. If you close all your private windows and then open another, it's a clean slate. (At least for me.)

I can see why they do this but it is actually not what I expected. I'd expect all windows to have their own set of cookies and credentials and for all tabs associated with a window to share them.

Are you logged in in a private window? I use the setting "Always use private browsing mode" in FF52, so all of my windows are in private mode, but whenever I open a new (private) window, I'm still logged in. I suspect you'd get the same behaviour with the default settings, and opening two new private windows.

FF53 on Ubuntu, stock settings.

And yes, you're right if I log in on either one of two private windows then the other one is also logged in. That's actually a bug in FF afaic.