But at the manager, low-level director or employee level, there's no responsibility.
From a management perspective, if it surprises you that someone leaves, it's because you weren't in touch with them.
At the moment you tell employees that it's their responsibility to tell you about someone who's job hunting, you will create a culture of mistrust.
It means doing that day-to-day boring management task of staying in touch with people.
If you think you'll curry favor, you are mistaken. The boss now knows what kind of character you have.