The only difference is which side of the visible blade the cut is.
You put a couple fingers through the bottom loop. Most people don't pull those straight upward, but pull a bit toward their hand. This pushes the blade to the left as it closes. Likewise with the thumb you push away a bit, which pushes the blade to the right. Thus, both blades are pushed against each other.
Sure, you could balance the scissors perfectly and only go straight up/down. But this won't work very well when cutting something difficult since there are no forces balancing each other out. It also won't work with looser scissors (every pair in elementary school) ad the blades will remain apart from each other.
I think there must be more to the left/right handed thing, because I'm effectively right-handed in everything I do except writing. Even in novel tasks I haven't done before: I'll try first left handed, it doesn't work, then I try again right handed and it works fine. But when it comes to writing specifically, I'm very clearly a lefty.
People tend to damage the left-handed scissors because they use them wrong: it bends the blades slightly apart. If you apply seemingly-excessive lateral force to the handles while closing the blades together, they work fine. (Also, new, undamaged pairs work fine.)
You do have to hold left-handed scissors at a slightly different angle to cut, though. If you're used to right-handed scissors, it's a difficult adjustment to make.
When you use left handed scissors with your right hand, you apply some force to the blades that moves them slightly apart, while with right-handed ones, you force them together tightly. You need to actively work against that.
Funnily enough, you can also see well if you cut with right-handed scissors on your left side, but with the paper held above your eye-line.