If it's their own car, all of it. I've had to work hard for everything I have and it's made me appreciate it that much more, and my kids will have the same opportunity. When they're old enough, they're free to get a job or do their own entrepreneurial activities. It's their money, they can do with it what they want. Cars are kind of like pets in that you can't just buy it and that's it -- recurring costs have to be figured into the plan to own a car. There's always a bicycle if one fails to plan and ends up with a car that can't be on the road for a while for whatever reason.
For the family car -- I like the way my mom did it, and will probably do it the same. My sister and I didn't have the family car often at all, but I did borrow it on several occasions. When I did, I just had to fill up the gas tank before returning it. I may only use a gallon and it needs 5, but that's how I put in my extra for all the other expenses of running the car. It was almost exclusively my mom's car, so it seemed the fairest way of sharing the costs. Obviously, if it's used by more people on a fairly balanced basis then percentages of total costs probably need to be figured out.