Everything traversed their network. It was a bonkers architecture that would not fly today. The other thing about that obscurity is it enabled all sorts of weird use cases. Because the devices were identified to the BlackBerry network, you could message without user assignment.
It was common for corporate and political people to use them for unaccountable, compartmentalized communications. You could build ad hoc networks of people without there a record of who was who, and periodically reshuffle the devices to add and remove people. It was basically Nextel DirectConnect for texting / “the wire” for corporate people.