Odoo Postgres Databases synchronization
I'm currently evaluating a similar setup and there are a couple of solutions.

What you're looking at here is basically a master-master replication scenario, where both databases are writable and changes in each are replicated to the other.

There is Bucardo, which is a postgresql-only replication system written in Perl and there are a number of tutorials for setting it up, check the Odoo community forums, too.

Then there's SymmetricDS, which does the same for a number of different databases and is written in Java.

Since your're dealing with master-master replication, you have to account for the occasion of both databases changing the same rows in the same table at the same time, so you'll have to configure conflict-resolution strategies (basically which change wins, but it can be a lot more involved like calling an external handler to figure out what to do).