When walking/running without a dedicated paved pedestrian path, it is highly advised to run AGAINST oncoming traffic.
When running on a paved pedestrian walkway, the side of the street is usually immaterial, but you should still observe the local pedestrian right of way – if most people walk to their right side, you should run on the right, and pass on the left.
The reason the side of the road doesn't matter when running on the sidewalk is simply that each pedestrian path usually corresponds to "both sides" of a roadway, and allows foot traffic in both directions.
Obviously, in cases with one-way walking paths (very rare), you should run in the same direction as indicated by signage/other pedestrians.
Do people run on the same side of the pavement as the side they drive cars on the road?