This is deadly woody nightshade, found growing through our fence from the neighbor's side. We've also encountered it and deadly nightshade at public parks and all over the alley. It's a highly fascinating and invasive plant.
Problem? Deadly nightshade is indeed deadly, or can be. Ingesting only a few berries can kill a small child. The foliage is also toxic. Woody nightshade also has toxins, though deaths from the plant are rare. Both can transfer disease to other members of the nightshade family, including garden tomatoes and some peppers.
Whenever we find woody or deadly nightshade, we pull it up, throw it in the trash, and wash hands thoroughly. We recommend you do the same.