Can 2 people claim head of house one has a child the other does not?

Not in this scenario. One dependent can only qualify one person for the head of household filing status. Even if there were 2 children, each person would have to be maintaining a household for one of the children to qualify.

Example 1: Two roommates split the rent and other household expenses. Each has a child and takes care of the expenses for their child. Each can qualify as head of household assuming they meet all other requirements.

Example 2: An unmarried couple has two children and live together. They live jointly and use their money jointly. They are not living as roommates. The person making the most money can qualify as head of household, because they would be maintaining more than half the costs of keeping up a home for a qualifying child. The other person qualifies as single. This is true regardless of who claims the children.