It's a finished basement..so I shouldn't finish the inside walls because there is already drywall on the garage side?The IRC code requires 1/2" rock on the garage side of the garage walls and any supporting structure (steel posts, stud walls, etc). If there is habitable space above the garage then 5/8" is required on the ceiling and any horizontal members such as steel beams. A lot of people spec 5/8" on the walls but the code does not require it.
In a single family residence under no circumstance should you have to have 5/8" (or 1/2" for that matter) on both sides of the wall separating the garage from the house. Just the garage side. The rock is intended as a fire membrane separation, not a true firewall that is designed to protect the structural elements from both sides of the wall.