Why is heavy precipitation from atmospheric rivers more likely to hit the West Coasts than the East Coasts?
Extratropical cyclones in the Northern Hemisphere rotate around the earth in the mid-latitudes in a mean westerly flow.  This means that, on average, the track of a storm between 30°-60°N moves from west to east.  Ahead of these extratropical cyclones is a strong frontal boundary with very powerful winds, sometimes up to hurricane strength near the boundary layer.  Moisture is advected (transported by wind) from the tropics (a very wet place, relatively speaking) by these powerful winds ahead of the extratropical cyclone. The moisture then creates a narrow channel of extremely moist air along the frontal boundary. Finally, the atmospheric river moves with the mean westerly flow as the frontal boundary/extratropical cyclone shifts east. Therefore, to answer your question, it is because extratropical cyclones on average move from west to east that ARs impact west coasts in the Northern Hemisphere.


Atmospheric River preceding Upper low in Mid-latitudes off of US West Coast

Another way to think of this is that tropical systems (those that develop between 0°-30°N) initially move from east to west in the easterlies.  Now ask yourself how often do the mid-latitude west coast's of northern hemisphere continents fall victim to category 3-4-5 hurricanes?  Caveat: remnants from weakening tropical depressions often move into the mid-latitudes before impacting these regions.


Track of tropical cyclones in the opposite direction of Atmospheric Rivers