Grounds and buildings shared by multiple owners

There are two solutions that come into my mind:

• Areas: Instead of adding the school and hospital tags to the corresponding buildings you draw two areas. The areas should represent the outline of the school and the hospital, respectively. Parts of the areas will overlap which is fine in your case. Now just add the relevant tags to the areas (amenity, address etc.). The school wiki page has a nice example explaining how to create such an area.
• Site relation: Instead of adding the school and hospital tags to the corresponding buildings you create two relations, one for the school and one for the hospital. Both relations will be tagged with type=site. Additionally they contain the school and hospital tags, respectively. Then add all the buildings belonging to the school (and possibly shared with the hospital) to the school site relation, and all the buildings belonging to the hospital (and possibly shared with the school) to the hospital site relation.