I got surgery done in the Mater Misericordiae hospital before. It is a huge Victorian era building that expanded into new modern office blocks. The hospital used to be run by the church, but now, directly because of Vatican 2 ( the downfall in vocations, and the new ideology saying that lay people should take over church functions) the hospital is run by ordinary people. I think the church just owns the building, but the hospital is funded by the government.
There is also a Mater private hospital not far from it which is not funded by Gov and that was never run by the church. for rich only.