The dogs that use it most successfully are dogs that were socialized very young and had good experiences as they visited. Depend upon and learn from the dog savvy people to see indications of an impending squabble. Redirecting the dog's attention at exactly the right moment can make all the difference. Your tone of voice and your body language will also make a big difference. Some are public, some are private.
Insurance can be a problem for private parks.
Public resistance to spending taxpayer money is a problem for public parks. Locally we are trying to form a group to promote a dog park. They are concerned, rightly, about spreading disease if it isn't picked up and then how to enforce pick up rules. Someone provides pickup bags. Usually it is the user group supporting the park.