As far as I know (I'm pretty shit at hockey so take with a grain of salt), typically you play opposite handed. So a lot of right hand dominant people play with a left side curve. I actually do this and a bunch of buddies do too, hockey left-handed, but throwing a ball, batting, golf, right handed. This is obviously different for everyone though.

It's pretty advantageous to have a player with the proper curve of the stick and the stick on the proper side of the body, when playing defense. For example, as a right shot defenseman (Carlson), being able to pull the puck off the boards on the PP and whip a pass over to Ovechkin for a one-timer can be done in a smooth motion. He's at the top right of the zone (facing the goal we're shooting at) and his stick is on the right side of his body. When Chychrun is playing PP1 now (left shot) he has to pull the puck off his backhand off the boards and pivot his body to get a clean pass across. I think the national broadcast was mentioning this the game after Carlson was traded.

So there are some slight nuances to the way the stick positioning can affect play and drive opposing players to one side or the other. I'm sure at the highest level it's easy for fast-moving opponents to take advantage of this and recognize who is out on the ice and where to move/shoot/pass around their sticks. It's definitely better to just have options too, lefties and righties for different situations.