The buyer is responsible. On our mortgages, the lender did require a home inspection as part of the paperwork, but we had to pay for it. If the home had failed the inspection in some major way, they probably would not have given us a loan.
I suggest you get recommendations for home inspectors from people you know. On this house, we took the recommendation of the real estate agent. What a mistake! There were plenty of things wrong with this house that the inspector did not catch. He seemed great but I had to get a separate inspector for the pool. I was not told but found out later, he did not inspect the chimney and we needed to get a different inspector for that. He did not go on the roof and check it out.
Make sure you are present for the inspection and take a camera and paper with you so you can take pictures and notes along the way.