This is something that comes up a lot when I'm teaching, and it's an important issue for drilling. Should you train everything both sides or not?
I definitely believe not.
Now, some things you do have to train both sides. If you can't bridge and shrimp escape both sides you are in big trouble. Same goes for any escape, especially when it comes to submissions. Generally, escape options are highly limited from any position (because so much of your body/movement is being controlled) so you should know them all well from both sides.
...but in attacking positions you always have a wide range of options and variations on them all. So if you're in a position where you're in control you should do what feels natural. That might mean you do drill something both sides, but if you find you really suck at something on one side, or you know you would never do it that side, spend the time drilling it on the good side. Why waste 50% of your time training below par?
If you're novice enough that when you drill something one side you just can't get it, but the other side you're ok, then getting really good on one side and rubbish on the other is definitely going to benefit you more than being ok on one side and still rubbish on the other.
If you're more experienced, so you can do a movement both sides but it feels much more natural on one, then as long as you have good attacking options from each position both sides, it doesn't have to be the same one. When you are fighting hard you will attack with what is natural to you.
By the time you get to an advanced level, you will pretty much be able to do everything both sides with relative ease as far as drilling goes, or at worst not take long to get the hang of it... but you'll still attack what is natural when fighting hard :)