I don't think it's right at the edge, though, it's in the middle of the yard. So either the dog is loose or being let loose onto the yard (and there are leash laws in the city).
The dog poop sign is a good idea, it's just that my husband needed to head over now and didn't have access to any (but we did have a No Trespassing sign on hand). If it continues, we could try to look for something like that. I know that when we had a dog, we certainly *did* stop her from going where she wanted to go if it was on somebody else's property (and she was an 80-pound black Lab, not a little dog). We'd pull her leash if she stopped and tried to squat, or, if that failed, we'd give her a shove so that it was too uncomfortable. She was allowed to poop on the neighborhood field (permitted as long as you bagged it afterwards, which we did). So I don't think "you can't stop them" is an excuse, although picking up afterwards seems like the *least* somebody could do.
I'm going to really hope that it's just somebody who figures, hey, biological waste, unowned property, no harm no foul, and that they'll stop when they see the No Trespassing sign right where they enter. Since we won't be living there ourselves, we can't keep an eye out to see who it is, but if they don't stop by the time renters move in, maybe the renters will figure it out.