With shielded twisted pair cable, do you ground one end, both ends, or neither ends of the shield
If both devices have their own ground reference then you should only attach the shield at one end. This way you do not produce what is called a ground loop.

When you do attach it I would attach it to the chassis as this is the return path you want noise to take, the chassis to earth, if you attach the device that I assume is inside the metallic case and then that attaches out to the earth reference you are bring noise inside the box that you do not need to.

If one device is operating lets say PoE and has no earth shield reference you want its shielding/chassis connected to the shield to act as its connection to earth for noise mitigation.