Wow, they've finally done it folks! In Firefox 20, making a private tab/window now works WITHOUT having to close your existing session like how Chrome's incognito window and Internet Explorer's InPrivate browsing.
Seriously, it should have been like this from the start. Opera's still one notch above the two quarrelling rivals though as it supports private tabs which you can mix with your existing non-private tabs in the same window.
Ready for some better private browsing? Enjoy!