Well, why don't we just test it for ourselves instead of wondering?
Let's see here: login to a site, open a new private browsing window (Chrome), load the site, am I logged in? Nope. Cookies from your normal browsing are not transmitted in private browsing mode.
Let's see here: login to a site, open a new private browsing window (Chrome), load the site, am I logged in? Nope. Cookies from your normal browsing are not transmitted in private browsing mode.
Which is not to say that private browsing is actually as private as you may think it is (see http://homepages.cs.ncl.ac.uk/feng.hao/files/DPM13.pdf).