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. They raise the money through donations or fundraisers. The user group tries to have someone present during the most used times. That someone spends a lot of time putting on a smile and educating. With time and effort and encouragement visitors learn that they all have a stake in seeing that the park is kept clean.