Does a small child on an international flight have a right to check in and carry on his or her own luggage?

yes but if you what it on the airplane it has to be a "carry on" suitcase or a bag.If you paid for a ticket for the child, then he or she has the right to have just as much luggage as any other paying customer.

Of course they have the right to check in or carry on luggage! This is usually limited to two carry-on bags in the United States. The same rules that apply to adults also apply to small children regarding cargo luggage. Tell the child not to forget their headphones; it's a long flight!