There is a real dog park at the east end of E. 3rd Avenue in San Mateo. There is also a park, as well as a path for bike riding and walking.
You can walk a dog and San Mateo Central Park. They have plastic bag dispensers in case you forget yours. There is a tiny fenced off area to let your dog free. It's so small that it's really only useful for small dogs. Aside from that small area, you always have to have a leash on.
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