My understanding:
If you want to fly through an ATZ you must contact the airfield and, if ATC, have obtained permission (Rules of the Air - Rule 39).
Assuming you are not in controlled airspace and you stay outside the ATZ (i.e. normally over 2000' AAL) you can do what you like and you don't need to contact anyone or obtain permission to overfly an airfield. However I think common sense dictates that if the airfield is manned you should at least let them know you are there and make your intentions known.
My policy if I am on a radar service and approaching an airfield just above the ATZ (say within 1000') I will request a switch to the airfield frequency to make contact, then go back to the radar once I have finished with them (either because the transit is complete, or if they tell me there is no traffic around).
If the transit is much higher (e.g. 4000' or more AAL) I think you are better on the radar where they can see other traffic anyway. If you are not on a radar service there is no harm in letting the airfield know, but they are not likely to be interested if you are well above the ATZ.