For which side they play, especially defense, the issue is quite simple. The North american style has significant board play (more on that later). Imagine you are a d man and you go into the boards. If you are a Rightie on the right side, when you extend your stick forward your body is open up the ice, your forehand shot is up the ice, your eyesight is up the ice. To move the puck forward you have a significant advantage.
Not take a leftie on the same side. His body is open to behind his net. His eyes are looking towards his own end/net. His forehand shot is back towards his end. He can try and shoot it around the back of his own net, or try a somewhat blind backhand to get it out of his own zone or to hold it in in the offensive end. A disdavantage.
Now you often hear its opposite in Europe, and that is quite true. Why? The much larger space on the european ice means the boards are a lot further away, and there is a lot less chasing of players by defensive players along the boards. European end play often looks more like an NHL power play (where a lot of players also turn to their offsides) where the defense looks more like a zone and they are quite willing to let a player hang out around the boards rather than get drawn out of position, especially with so much ice and a lot more/larger potential passing lanes.
Of course the opposite is true for offense, having the puck towards the middle has distinct advantages too. But in the more crowded NHL sized rink the middle generally gets pretty crowded, so the advantage is limited.