Three situations with which I have experience may help you.

1) The Sepulveda Basin/Apollo Field site in Los Angeles, CA. It is on land owned and operated by the City of Los Angeles Recreation and Parks Department. They allow both AMA and non-AMA pilots to fly. If you belong to one of the several clubs that use the site then you are required to be an AMA member. There is no requirement to demonstrate proof of insurance if you are a non-AMA member. They do ask the AMA clubs to sort of police the field rules and have posted the AMA safety code and stated taht pilots should abide by that. Many do and some do not and there is little that can be done under that arrangement.

2) The Whittier Narrows field is located in the Whittier Narrows park operated by the Los Angeles County Parks Department. They do require AMA membership. They also require NAMBA membership of those that use the lake in the same park for RC boating.

3) The City of Costa Mesa requires a permit to fly electrics or gliders in their Fairview Park site. To obtain the permit you must either show your AMA card or proof of liability insurance in the amount of $500,000 (I believe this is the amount). You do not have to be an AMA member, but you do have to prove you have insurance.

So there are three different situations all operating happily on public lands. I might add that all three of these situations are on sites with unrestricted access by the public and not under the control of a single club by way of a lease agreement. A fourth example is Prado Regional Park in Chino, CA. There is a very nice flying site but it is operated under lease to a club. That club controls access and requires AMA membership. In the same park there is a lake where you can run RC boats. No club controls that site and the Parks Department absolutely requires NAMBA membership to run boats there. SO they go both ways!!