It's hard to say, really.
I've always grown up around pets, be it my own or another family member's, and that's been long since before I could walk. People would leave me alone for a minute and I actually remember my mom having pictures of when they would return and the animal would be laying next to me or cradling me or what have ya.
So, given the theory that Pokemon are much more intelligent than our own animals, I'd say that a very young child actually having a Pokemon of their own wouldn't be something too obscure to imagine. Especially if the Pokemon itself is a decent bit older than the child, then I'm sure it understands the difference between an adult human and a younger human, especially if it were the case of the adult's Pokemon's offspring being given to the child, where the mother of that Pokemon would be there to oversee the young Pokemon's powers and abilities in check around the human child, and the human parents to keep their own child in check as to be gentle and kind to the Pokemon.
As for journeying out on your own with Pokemon, 10 is just way too young. Like, way too young. At nine years old, a child's brain has JUST been able to accept the fact that a hotdog cut into tiny pieces is still the same amount of food as a whole hotdog, where as a child any younger still sees the tinier pieces as being more food, because there is a larger amount of it. A child who has JUST learned that not even a year ago is just way too young to be given a giant fire-breathing dragon and told, "Go out on your own and be a Trainer."
To me, 17 or 18 is about the age I'd let my child venture off with a Pokemon. Much like how many parents do in our world, I feel that 18 is ready to start making decisions of their own about where they should live, what they should do, and where they should travel to.