Does the other driver (or their insurer) have to contact me before they repair the other driver’s car?
No. There is no requirement that they contact you before repairing the car, or let you assess the car, or give you an opportunity to repair the car.
The insured is entitled to have his or her car repaired, and then seek the reasonable cost of this from the at fault party.
The first contact you get will often be a demand for payment.