ORAM over Third Party DB
ORAM  can be built on a multi-party scenario, where the data belongs to various parties using Secure Multiparty Computation. Recent research calls this approach ORAM for Secure Computation (SC-ORAM).   

Given that writing and reading are required, it is my believe this would be the best approach.

In the typical SC-ORAM setting, several parties store information on remote server that holds the ORAM, and use SFE and general 2PC to execute ORAM tasks.  

There are ways you can get rid of the Server however, Keller and Scholes propose the adaptation of Path ORAM, among other methods under a setting where the DB is secretly shared among the participants. Their results could be adapted to any method using ORAM and MPC. 

What may be the most relevant of all, are the results by Faber et al, show how to build such setting under an MPC framework that allow for a secret shared database between 3 distrustful parties. 

A short post about their results including a nice video with detailed explanations can be found here.