[gnicholas]

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.)

[jacquesm]

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.