The way OHDSI is set up is that the data are private, and everything else is public, open and transparent. This is actually the strength: Moving data out of their institutions is an extremely cumbersome job requiring formal data use agreements and extensive data security frameworks. With the OMOP CDM this is not necessary: You can query data remotely, meaning, you can develop the queries that will "run many different types of analyses", send them to data owners and get the answer, without having access yourself. It's like eating the cake without having one!
What you want to do is this: Post your research questions on Github. Folks will make their data available. If not, prod them a little. We'll help you.