The AMA is a large club, which offers insurance and a set of standard rules for its members and affiliated clubs. Think "AOPA for RC flyers". Again, they have no regulatory authority. The local RC club has absolutely no say in what you do or don't do, other than flying at their field.
Most of them know that, but in most clubs you're going to find a few people who want to try and assert some authority and become self-appointed cops.
If they're concerned about radio interference from your flying, that's covered by FCC regulations (hint: neither they nor you have any protection from interference). As for flying objects, again... FAA jurisdiction.
For the FAA's position, Google and faa.gov are your friends. From the FAA web site:
Model Aircraft
Recreational use of airspace by model aircraft is covered by FAA Advisory Circular 91-57, which generally limits operations to below 400 feet above ground level and away from airports and air traffic. In 2007, the FAA clarified that AC 91-57 only applies to modelers, and specifically excludes individuals or companies flying model aircraft for business purposes.
The FAA guidance is available at: http://www.faa.gov/documentLibrary/media/Advisory_Circular/91-57.pdf