OpenStreetMap will always accept building names if they're correct and their addition is legal; removing such information could be interpreted as vandalism by some. If you remove the building names then it is quite possible someone else adds them back in, and you can't tidy up the area forever. The only way to get your data "officially removed" from OSM would be if you could show that it was copyrighted information (e.g. copyied copied from another map) or publishing it was violating the law (e.g. names of occupants), where "the law" would largely mean "English law" since that's where OSM is based. (We don't stop mapping China just because it is against Chinese law.)
You can of course reach out to the mappers (and in fact before you remove anything you should reach out to whoever added the building names in the first place) and try to convince them that adding in the building names is not a good idea. If you are with the school administration and the mappers in question should turn out to be employees or students then that would probably make for a more convincing argument on your side. But in the long run it is unlikely that you will be able to keep this information out of OpenStreetMap.
(NB in many cases where OSM has been contacted about sensitive information that shouldn't be on the map, a quick search with the search engine of your choice revealed that the information was already public.)
