Technically they aren't putting their arrows on the wrong side of the bow. It's simply two different styles for drawing a bow. The more well known Mediterranean draw, which requires you to place the arrow on the left side of the bow (right side if you're a left handed archer), and you pull the string back using either your index and middle finger (two finger Mediterranean release), or your index, middle finger, and ring finger (three finger Mediterranean release). The way in which they place their arrows in the game is the Mongolian draw which is widely used throughout Asia. The arrow is placed on the right side of the bow (left side if a left handed archer), and the string is drawn with the thumb. Because of the way the bow is drawn, and the way the arrow is held, the arrow actually stays in place because the arrow acts like a lever and the force that is placed on the back of it by the webbing of your hand between your thumb and index finger, presses the front of the arrow to the side of the bow and prevents it from moving around.
The start of this video will show a demonstration of what I mean. He is shooting left handed, and therefore the arrow is on the left side of the bow. If he were using the Mediterranean release, the arrow would fall off or flop around. But because he is using the Mongolian release (used in China, Japan, Korea, Mongolia, etc.) the arrow stays put.
