The pipe itself may very well be private property (i.e. the lot owner's responsibility to maintain and repair).
However, I think repairing the damage caused by the water from the pipe to Common Property is still the OC's responsibility.
The lot owner's contents insurance might cover repairs to the pipe (mine does). But the owner might not have contents insurance as it's not compulsory. The OC's compulsory building insurance should cover repairs to the Common Property at least. It then becomes the OC's insurance company's decision to decide to try to recover these costs from the lot owner.
Our compulsory building insurance covers a lot more than just Common Property – some of lot contents are covered too, e.g: “Proprietor's fixtures, fittings and improvements which form part of the building”.