Best way to manage common contacts between 2 CiviCRM databases?
I am by no means an expert or experienced with multi-site or multi-org implementations but this is my take on it.

If the two Orgs were going to be perpetually involved with each other then it would make sense to setup one Civicrm instance and share the database. Utilizing the Multi-site and Multi-org settings found here.

However, since the two Orgs will not be perpetually involved that doesn't really make sense (at least to me). I would, if I were implementing this instance, setup Civi for each Org separately. I would obtain an export (hoping Org A allows it) of their contacts and import those contacts into the Org B's Civi install for an initial load. That way you would start out with identical contact databases. You could occasionally obtain an export from Civi OrgA's contacts and import into Org B's civi install utilizing a de-dupe rule. 

You could also, if you have a developer or want to pay for it, create an extension or script that would sync the contact databases at regular intervals.