The alert dogs I have encountered alert their handler, maybe by nudging with their nose. Then the handler knows to test their blood sugar or sit down to avoid falling in a seizure or whatever. They aren’t alerting other people.
There is also a difference between a psychiatric service dog, which is trained to do specific behaviors that interrupt panic or ptsd symptoms, and an emotional support dog which just makes you feel less anxious by being present. The former can go anywhere that other service animals go. The latter is only protected in specific situations, the big ones being housing and planes. So if someone tells you their purse dog makes them feel less anxious that dog isn’t entitled to be in the grocery store.