Who is supposed to entertain my young child at someone else’s home?

You are responsible for keeping your child out of trouble. If the host family has children who are willing to include yours, that is fine, but do not just tell your child to run off and play. Americans are somewhat more strict with young children and you should follow the pattern by keeping your child as quiet and calmly entertained as possible. If it is an adult gathering, leave your children with a babysitter!