I regularly sell deactivated stens at some of the local gun shows, and get the sten/hand grenade story all the time. I usually stop the guys telling the story in the first sentence and give them the list of reasons why it would not work. Claven has mentioned the biggest reason, and that is that while a sten may go off once if it did happen to land on it's butt, but on firing it would cock itself fully and that would be the end of it. And even if it were mechanically possible for it to empty it's mag, where does the rotational force come from that is going to make a sten spin round and round?
The most likely scenario of such an attempt is that one of your party would now be unarmed, while an opposing soldier would now have a spare. Besides, one would expect that the basic military training a soldier would have received generally ingrained upon him the importance of protecting his weapon.
I have also heard the same story modernized to include the C1 (sterling) SMG.