These things are indeed possible while maintaining fully semantically secure encryption. Recent, mostly theoretical work shows that this is possible using fully homomorphic encryption. The basic idea is, the client can encrypt its query, the server can process the encrypted query and produce an encrypted result, and send this back to the client. It sounds impossible, but it isn’t! Very cool stuff. There are actually also some practical implementations that work… so it’s gradually exiting the “theoretical only” stage.

MongoDB is very short on details, and I suspect they do something worse than homomorphic encryption, that does indeed make some kind of compromise between privacy and convenience.