Some parts of Los Angeles are quite dog friendly. You might want to look here for example for facilities close to Beverly Hills --
DogFriendly.com's City Guide: Hollywood - West LA, California
If you check Dog Friendly, Los Angeles you'll probably see more listings.
You will find that the "on leash(lead)" requirement is virtually universal here except for dog parks. Frankly, although my two will stay right by me, I'd never let go of their leashes in public areas. That is just a safety and security issue for me.