Is it possible to construct dataset for only another party to consult? Is it possible to construct a dataset made of some kind of probabilistic encrypted indexes such that anyone can update the information contained in the set but only the one with a secret key could be able to retrieve the information for a given index without the need to retrieve the whole set?

For example, if I have a set of messages sent to me hosted remotely, like an e-email service, indexed by dates, is it possible to obfuscate the dates such that anyone can still send me a message and index it (by inserting into the set), but no one with access to the set can gain no information about the dates distribution except for the ones with the secret key? With the restriction that when I want to retrieve messages from a given date, I only need to receive those and no others.
Answering my own question. I believe it is possible. We can use the scheme by Boneh et al. described here. 

It uses pairing-based cryptography to be able to create a searchable, asymmetric, tagging scheme. In this scheme, queries are made through a trapdoor of the real key that we want to search, making it impossible for the set holder to retrieve information about the underlying queries.