I almost *always* go directly to the kid, esp. if the parent isn't there.
And I assume that they are nice people who wouldn't want to do something that upset someone or bugged them, etc. But that, because they are so young, they simply don't have the depth of knowledge and experience.
And so I don't say, "please use your inside voice," I say, "Could I ask you a favor? When someone is young, like you, their voice is really piercing--it carries really far and is louder to listen to that you think. So when you squeal, it actually hurts some people's ears. And it makes it hard for the grownups to listen to things. Can you try to remember not to squeal?"
This weekend, I stopped at kid (about 8?) at the church Christmas party who was walking across the room with a candy cane in his mouth. And I said, "Can I ask you a favor? I want to ask you not to walk with something in your mouth, and here's why. It really bugs all the grownups. All these grownups have really good imaginations--you think kids have all the imagination? Ha! And when they see the thing in your mouth, they immediately start picturing you falling, or someone waving their arm and bumping the candy cane, and then it scrapes the roof of your mouth, or punctures it, or maybe even it goes up in your BRAIN, or it knocks a tooth loose, or even if it just makes you cry! And then they get all antsy. So it would be nice of you to take it easy on them and not walk with something in your mouth. And especially don't RUN!"
He said, "OK," and took it out and walked off.
In the case of the kid kicking the airline seat, even if the parent is right there, I directly say, "I wonder if you don't realize exactly how thin these airline seats are--when you kick the seat, I can feel it almost as strong as if you were actually kicking me. And it shakes my seat a lot, which is really bugging me. So I wanted to make sure you knew, and ask you if you'd stop kicking the seat."
And I smile and use my friendly voice.