My bathroom fan brings air into the bathroom; should it?

From what you describe it seems unlikely that the vent fan is actually installed to blow backwards. So it would be much more likely that the trouble you are encountering could be one of:

1. The back draft flapper may be stuck in a shut position. This is a device to close off the vent for air flow trying to come in from the vent pipe. Sometimes they are built right into fan housing where the vent pipe attaches. Other times the flapper will be part of the roof or wall vent where the vent pipe exhausts to the outside. In some cases there could be flappers in both locations.
2. The vent pipe from the from the fan to the outside may be filled up with dust, dirt or other debris. This could be from years of accumulation or debris blown in or carried in from the outside vent opening.
3. The vent pipe may be damaged, kinked or flattened. This could happen when workers were careless or for other various reasons due to remolding, adding insulation or re-roofing.

In all three instances there can be blockage of the airflow that the fan is trying to expel from the room. For a significant enough blockage some of the air flow will try to return back into the room around the periphery of the fan blades. This is likely what you are noticing when you held the paper up to the vent grille. It is often the case that a fan trying to blow into a blockage will generate a lot more noise than one that has a free flowing path for the air that the fan is moving.

The fix is going to require careful inspection of the whole vent system from the fan housing all the way through the vent pipe and out to the place where the pipe terminates the roof or wall of the building.