Maybe you have a different grass than I have in my yard, but the coarse zoyzia grass in my yard hasn't "spotted" or "browned" because of dog urine in the 2 1/2 years I've been watching my neighbors walk their dogs down my street.
When I first moved here I was irritated about the dog walkers (I'm a cat guy) who "allowed" their pooches to relieve themselves in my yard. I've even been "tactfully reminded" that the first 10 ft. is not technically mine. But after a while I noticed that the grass wasn't being affected, unlike my rye grass mixture lawn did up in NJ.
So now I just say hello to my neighbors from my street facing lanai (that at least let's them know I see them) and refrain from stressing out.
(Except for the one guy I saw standing with his dog in the middle of my large corner yard. "Hey, this isn't a public park buddy!!")
