As you know, Firefox 20 has a whole new way of doing private browsing ,so some bugs are to be expected.

I didn't think it was possible to open a non-private tab in a private window. When I try to drag tabs between a regular window and a private window, Firefox will not move them.

If you mean that you've discovered a link that launches a new non-private window from a private window, perhaps this is a link inside the Flash plugin itself? Firefox might be failing to detect that those originated in a private window. If you can spot a pattern to it, that would be really helpful in tracking it down.

Many thanks for your help. The problem I refer it's easier than the approach you suggest in the second part of your answer. Let me put in a very simple way what happens. I start firefox in the normal way, non private session and suppose I have two or three tabs opened. Then I decide to start a private browsing, again with two or three tabs. Both browsing modes are opened as we can see overing the mouse on the firefox icon and it's possible to choose one of them clicking on the image over the icon. What happens frequently is when in private browsing clicking to see a video clip, for example in youtube, instead of opening the video clip and continuing in private browsing, the existing non private session opens exactly as it was. The video clip opens and the private browsing session continues. If you over the mouse on the firefox icon you can see the two sessions namely the private with the video clip playing. Just click on it and you change to the private session. So, there are not non private tabs opened in the private window nor any link that launch a new window. Both sessions stay as they where it's just a small inconvenience that to get the private session you need to over the mouse and click on it. As you suggest it seems more a small bug due to the new way of doing private browsing. Kind regards

I think I was distracted by the private/non-private aspect...

There is a bug with the Flash plugin (version 11.3 and higher) on Windows 7, even in all non-private browsing, where opening a page with Flash media causes that window to move behind another Firefox window -- if no other pages have used Flash within the previous 3 minutes. You can read more about that bug in this thread: Opening New Windows and Shockwave Flash.

It's possible that the isolation between private and non-private windows has made this problem worse, if each new window gets its own load of Flash, but I haven't experimented with it just yet to see whether that seems to be the case.

Many thanks jscher That explains why I never had this problem with chrome, although as said I use this browser only about 5% of the time, as chrome doesn't use the Adobe flash plugin. Hope they will solve this problem soon although the inconvenience is really small at least in my case. Once more again thanks and accept my best regards.