How can I unclog a passive bathroom air vent?
The vent doesn't need to be unclogged. You stated that paper is held to the register. That scenario shows that air flows through the vent unrestricted. The probable reason it might appear to be blocked (when the window is closed) is you reside in a well sealed building that has few drafts. With out going into detail about air flow; the reason you detect air movement when the window is opened is due to air flow from outside having a unrestricted passage through your vent. When the window is closed the air currents have to detour around the building walls.