My background has been with general contractors, although I'm doing engineering work now, so my position may be colored by that perspective. I maintain it is the Designer-of-Record's responsibility. Now, he may delegate that responsibility, but ultimately the buck stops with him.
If the DOR is not confident in doing the actual connection design and expects the stone or precast panel supplier to perform the design, then the specifications should be absolutely clear that that is what is required. In the end, the DOR still has the ultimate responsibility for the overall design of the building.
This is similar to any component design of structural steel, precast concrete or wood trusses. The DOR specifies the design requirements and loads and then delegates the final detailed design to the component specialist.