Contractor HSE: how far does your duty of care actually reach?

A contractor arrives with their own method statement, their own competence records and their own safety culture. You are responsible for the site they are working on, and in most jurisdictions for a good deal of what happens to them while they are on it. The practical questions are where to check and where to trust. Do you re-verify competence for the people who turn up, or accept the contractor assurance? Do you audit their risk assessment or only its output? And what do you do when a long-standing contractor fails a check at the gate and the shutdown starts on Monday?