I was walking on (upper) Market St when a woman, sitting on a stoop, screamed out for help. I turned, saw her and then saw a flight of stairs behind her. She was wailing as if she'd just fallen down the stairs. I ran over to her, asked if she was ok (she was lucid so I figured she could tell me what had happened), but she just stared up at me screaming. I called 911 and told them what I thought had happened. The fire department, big rig and all, pulled up in about 2 minutes (station close by, but amazing response time, really) and the moment the first fireman hopped out of the truck you could just see the exasperation on his face. He made no urgent rush over to the woman. He just looked at me and asked if that was the woman and explained that they'd already been out to see her earlier that day, and many times in the past. I felt terrible, but of course had no way of knowing. Fireman just patted me on the back and said I did the right thing.
The moral (from an unnecessarily long story): it's almost impossible to know what's an emergency in these situations.
My thought (reading this on HN where I ignorantly think tech can solve everything) is to say that 911 calls in SF should include a picture of the person, but that's not sensible, of course, because seconds can sometimes literally be the difference between life and death.