No, you don't. One cinder database is all you need.

Although note, that cinder-volume processes do NOT connect to the database directly. Instead, they communicate with the cinder process running on the controller node (I believe with the cinder-scheduler), which then communicates with the database on their behalf. You typically need only one cinder-api(controller node), one cinder-scheduler (controller node), and multiple cinder-volume (storage node) processes running.