How to play against little kids? Say, in an age range of 7-12 years. One of my friends holds the opinion that you should trounce them at every opportunity, so that they can learn from their mistakes. Another says that you should allow them to win sometimes, so they won't lose interest/stay encouraged to play chess. Perhaps the answer lies somewhere in the middle. What do you guys think is appropriate in such circumstances?

(Context: I volunteer at a local elementary school to play/teach kids occasionally. Some are highly motivated, others seemed to be forced to be there by their parents.)
I think the best approach is what I call live annotated chess.  In this case you basically play normally but provide live feedback on the student's position and why you are doing the things you are doing.  This allows you to understand where the student needs additional help and allows them to learn from you.  

This approach also allows you to adjust your play according to student needs.  If the student is making a lot of careless errors and doesn't seem too into it, you can tone down the annotation and your game and let them play more for fun.  If the student seems really motivated in a given game, you can introduce more challenge by upping your game and your discussion.    

You can even do this within a game if you sense the student getting frustrated or bored.

I would also say that not all chess needs to be played by "chess rules" kids at that age get a kick out of manipulating the world around them. Let them make up new games that still teach chess principles but are not so rigid.  Let the bishops move like queens, or allow them to win back material for consecutive captures, etc...  I don't think these types of things teach bad practice because they let the student experiment with the "world" and understand how it reacts.