Yes, every structure with more than a multi wire branch circuit has to be grounded with an electrode specifically for lightning and over-voltage;
see 250.4(A)(1) this is it?s purpose & 250.32. The rod has nothing to do with the neutral at the separate structure though.
In summary [essence]:
- One service, one service disconnect, one electrode at service disconnect, & bonded common neutral.
- Separate building, separate building disconnect, separate ground electrode, & separate neutral from equipment ground.
The electrodes must bond/connect past the service and this is done via the equipment-grounding conductor (sized per 250.122), causing A, B, N, GND.