How often do graduate students help connect professors with each other?
It happens, but rarely through co-advising.  In my experience, co-advisors usually know each other before the student enters the picture, and the student has little effect on how much the faculty collaborate.

More often, students create new connections by becoming active independent researchers and working with a diverse set of collaborators.  The various actors' roles as "student" and "professor" are simply irrelevant.

(I'm speaking for theoretical computer science, where PhD students are expected to become full-fledged independent researchers before they graduate.  Your mileage may vary.)