It is really hard. Like a previous poster, I have just moved and am confronting this on a scale I've never imagined. I moved from a city on the west coast where that kind of thing is just really unacceptable and wouldn't be tolerated in public, to a city on the east coast where everyone is well invested in the "mind your own business" approach which has allowed people to be awful.
I am a mandatory reporter in my state, so I have to report any abuse I witness or have reasonable cause to believe is occurring. When I don't really have any information on the person, it isn't possible to do much with that. I know some mandatory reporters try to chat it up with someone to find out names or child's school or where they are going or some kind of identifying info. However, I would second that it is a good idea to call the police if you are seeing physical stuff manifesting right in front of you.
That said, yesterday outside my office I was hearing a baby crying and a dad (from the temp shelter currently in the building) repeatedly yelling at the baby to just "shut up!" I was getting so anxious just listening. I finally went out to the room where they were, and I just observed for a couple minutes. Here is what I saw:
The mother was sitting on some stairs, and the baby was plopped down (sitting) on the floor about three feet from her. There were many bigger kids playing around her, and there was an (unpredictable) ball flying everywhere which helped make the baby feel vulnerable. It looked a couple of times like she was trying to scoot to her mom but couldn't get the mobility. That seemed to contribute to her frustation, especially since she seemed like she was needing some comfort. Another baby, who might have been her twin, was crawling around as well. The crying baby was obviously sick. She kept sneezing and all this snot would fly out all over her face, which would make the crying worse. Then her dad would come up behind her and without letting her know he was there, he would reach around and wipe her nose. She was taken by surprise and also hated getting wiped. Meanwhile he would yell at her "Oh, knock it off" when she responded with tears, and then he would proceed to pace around the room telling her to shut up and stop crying as he waited for the next sneeze and interacted with the other kids.
I went and knelt down beside the baby. I crouched as low down as I could so I was on her level and I looked at her and just tried to model what would have been a nurturing response. I said, "Oh sweetie! You sound really upset. What's the matter?" The mother continued to just sit there, and it seemed okay, so I rubbed/patted the baby's back lightly a couple times and said, "You sound so sick baby girl! I'm sorry. I am miserable when I am sick too."
This seemed to really click with the mom and she said, "I know! I've been telling her dad we need to take her to the doctor, but he doesn't want to." So when the dad came back over to wipe her nose again I just repeated, "She sounds so sick, poor thing!" He didn't yell after that, and after asking the mom how old the baby was and telling her she was a very sweet baby, she said thanks and then picked her baby up and went to talk to a friend. It seemed to have difussed the situation.
Sometimes I think ignoring the parents but engaging with the kid works well. It just breaks the tension and occupies both. Other times, I say things to my kids, loud enough for the offender to hear, that I think might help. For example I might have said, "Oh [3 y.o. ds], that baby is crying. She sounds so upset." And then ds pipes in, "What's the matter with her?" And I say, "I don't know, but it looks like she feels really sick. Isn't it crummy when you feel sick. I wonder what would make her feel better?" And then hopefully ds would pipe in that the baby might want to be cuddled or something like that.
Other times I say things to my kids, not really intending to intervene but more because I don't want my kids witnessing that stuff and thinking that it is okay, and it ends up being unintentionally overheard, which can make things better or worse depending on the person. For example, one time this woman was flipping out on her daughter in a parking lot, just screaming in her daughter's face. Sometimes going up and offering to help in situations like that is good, but in this case I got the feeling that wasn't going to fly. ds however, said, "What is she doing?" I told him she seemed really frustrated and was yelling, and I told him that yelling like that isn't okay and that the mommy should do something to calm herself down. I am not sure, but I think she overheard me, and she seemed to at least get a little more reasoned about what she was yelling rather than just hurling insults and calling her daughter names. I think it just reminded her that she was being heard. I've occassionally tried to gently point out reasons for a child's behavior that had become an inappropriate target of the abuse. For example, "He looks so tired! I remember doing that when I was tired when I was a kid too. Is it almost naptime?"
In extreme situations I've been known to say, "You should know that I'm a mandatory reporter" (not sure if you want to use it if you aren't, but maybe when you are desperate). Or just "whoah!" and then flashing an alarmed look their way to let them know, "I just saw that, and you were way out of line."
I gotta say though, like you, I often freeze up and regret it later. I totally empathize!
Though your post was to the OP, it really touched me. I learned soooooooooooo much from you tonight. These are very constructive ways to handle things, and I'd like to implement your strategies in my own life.
I do know what you mean about the mind your own business attitude. I really do have a hard time minding my business. I can't tell you how many times I called the police on people just to make sure things were okay. The county cops know me pretty well.