Well it can be both parties paying when necessary. End day it is you and your spouse child. Perhaps he has reason why the first child was in private hospital. If you wish to have your own child be in private hospital you can speak with your spouse to avoid misunderstanding or arguement. But either which of the hospital, as long as check up are done properly, gynae, nurses and service are tip top that's already good. But having said to my personal opinion, being in private can be much more expensive, some couples can't keep up with the expenses. Their only reason to take private can be better and prompt service or their gynae are located there.
Do discuss this once again with your spouse just to clear the air. Virtual hugs 🤗