I'm looking for (a) word(s) that is/are perceived to be child's language by adults, not words used by adults to describe children. What would be fine though are words used by adults when they are faking child's language, be it in playing with children or in mocking each other. Dialectal words from specific regions (i.e. Staffordshire, Big Bend Country, Shannon, Manitoba) would be very welcome!
The word most frequently used by far in American English by children to refer to other children is kid itself.
I met this kid at the playground today.
I'm sitting next to a new kid in class this week.
After that, gender-specific terms to use by a boy would be girl (to refer to a girl) or boy by a girl to refer to a boy. This would be to distinguish the gender because any anonymous child referenced would be assumed to be the same sex.
There are no built-in pejoratives. You'd have to add an adjective or use another noun: "That stupid kid" or "that dirty boy" or "idiot" (depending of course on what traits the speaker wanted to single out).
The one word practically never used by a child to refer to another child would be child. Same with children.
There's a bunch of new kids in my class. [not children]
Having had this question in the back of my mind for the better part of a day, I feel obliged to observe that "kid" is probably a bit less likely to be used when referring to a girl by another girl. Not that it couldn't be, but in American English, as in many languages, generic terms seem to have a default "maleness" about them. I don't want to get into the politics of that here, and do recognize that I say this only as an observer, not a proponent. But there it is. In the comments to this question, Brian observed a correspondence between "kid" and "guy" and I do endorse that similarity. I would extend it further to cover this kind of vague male undertone to the word, as I have done in the past. I think the polarity is possibly less with "kid" than with "guy," but I could be wrong.