With the feedback from Alex Faaborg on the status of Private Browsing in Firefox, and urged by the Incognito browsing mode in Chrome and Internet Explorer's InPrivate Browsing (and of course, Safari's Private Browsing mode), it seems that there's a chance to have this in Firefox 3.1. In case you don't already know, I had written a patch which added support for the Private Browsing mode a while back, but it never saw the light of the day. Now, I have a new patch which implements the new requirements nearly completely (the only part missing from it is disabling DOM Storage in private mode.
What happens when you run Firefox with this patch is, when you enter the private mode via the Private Browsing menu entry under the Tools menu, all of your logged sessions get invalidated, your whole cookie list is cleared, and the site permission controls in Page Info windows get disabled. When you're working in the private mode, no record of your browsing history is ever saved, all cookies are treated as session cookies, Firefox will not auto-fill the password forms for websites with saved passwords, and will not prompt you for saving the password on websites where you enter your password for the first time, and also will not save auto-complete entries for what you enter in the web forms or keep your downloaded files in the list of your downloads. This will make it very difficult for anyone using your PC to tell which websites you have visited during the private browsing session (although the owners of the websites will be able to track your browsing just like in the usual browsing mode). When you exit the Private Browsing mode, all browser services return to their normal operation, just like the moment you entered the private browsing mode.
At this point, we are looking for people willing to test this patch. For now, you should apply this patch on Firefox trunk and build it yourself, but hopefully we'll get builds for Windows, Mac OS X and Linux ready shortly so that people can test this new mode more easily. Feedback in form of comments on this post as well as the Private Browsing bug is much appreciated.