Everything about private browsing is described in this article: https://support.mozilla.org/en-US/kb/private-browsing-use-firefox-without-history

PB mode only means that Firefox doesn't store personal data from the session in the Firefox profile folder. It doesn't make a difference of how the website sees you and cookies work the same as in normal mode. You can use the Password Manager to fill in already stored passwords but you can't stores new passwords in Private Browsing mode. You should use a master password if you currently aren't using one. *https://support.mozilla.org/kb/use-master-password-protect-stored-logins