Should I have someone cut down my pine trees in exchange for wood  

I would make some conditions - perhaps that they would not be able to actually remove any of the wood until they have cleaned up the mess of the branches, leaves, etc. that is left when you cut trees - otherwise, someone might leave you a really big mess! I would also want to tell them that they can't drive through your yard when the ground is soft - you don't want them leaving big ruts in your yard.

I would make some conditions, too... but you might want to check with your homeowner's insurance to be sure you'll be covered if there's an accident while they are working. Other than that, it sounds like it would be a good deal for both of you.