Help with figuring out the airflow on a split-system heat pump?

I'm not sure which way the street is but that big hose is called your return duct. The arrow on the filter should be pointing towards the air handler and away from the return duct as it is slid into the filter housing. The arrow denotes air flow direction. The return air is doing just that, returning to the air handler therefore flowing in that direction. The dirt and gunk would be on the right side of the filter as the filters main job is to protect the coil in the air handler which is why you filter the air right before it gets to the air handler.