I'm working with Firefox 43.0.1 on Windows 7 Pro 64 bit. I just noticed an odd behavior, and I don't think Firefox used to behave this way.
I had a GMail tab open in private browsing mode, and then opened a new tab in that same window to go to YouTube. YouTube recognized who I was and asked which account I wanted to use to browse YouTube with. I promptly closed that tab, opened a new private browsing window and went to YouTube and it auto-logged me in as one of my accounts.
I was surprised by this behavior, though I'm not sure that I have previously tested it. I expected that private mode would not allow webpages/sites access to other cookies and such, for purposes of limiting tracking and whatnot. Even if it does not do this per tab, I expected each window to be pseudo-seperate. Therefore, I would expect the above scenario to not occur and YouTube to not know anything about me and not auto log me in.
How is Firefox supposed to manage tabs and windows in private browsing mode? Is Firefox supposed to allow websites access to the sort of data that knows if I am logged in or not? Are my expectations realistic? Am I expecting too much?
I found this Mozilla Wiki page explaining exactly what I expected, and if I am reading this right, it says that it was implemented circa Firefox 20.
I am also running several other privacy and security minded extensions, so maybe I previously experienced what i expected due to the combined effects of Firefox and the extensions.