By database I am assuming you mean P2P block files, right? Otherwise if you just fetch the server state, it is not truly private because the blockchain is still obtained server-side.
I was just using that terminology to introduce PIR, as it's commonly applied to database queries. As the blockchain is somewhat of a decentralized database, I believed it should be possible to adapt PIR for Bitcoin blockchain queries.
I've yet to check out the paper BlackHatCoiner sent, but at first thought, whatever they did should be possible to adapt to a light Bitcoin client scenario.
All it really needs is querying balances and transaction histories of addresses, as well as submitting transactions to the blockchain.
Not sure whether they have a feature for that, though - few block explorers do have a 'submit raw transaction' feature.
If the paper's authors provide an API to their block explorer, it may directly be usable for the application I had in mind.
I get that it has to be a "light" client, but with no central oracle to verify all the nodes' authenticity, you're still trusting that the database sump hasn't been tampered with before the connection opened (something that encryption cannot remedy).
If you are referring to the general limitation of SPV wallets (trusting the server providers); that's true, like it always is. A partial remedy would be having multiple servers and avoiding large amounts.