There's a slight pressure you put on the hinge when closing them. The handedness aligns the pressure to a closing motion laterally (keeping the blades together) as opposed to opening (pushing the cutting edges apart).
They do in fact make a difference. Trimming my beard with beard scissors requires a bit of a strange grip in my left hand to actually cut well when I switch sides.
Nail scissors can be quite the experience for a left-hander, instead of achieving the proper cutting/shearing motion it will just try to bend your nails due to the widened gap between the blades. A bit less painful but also annoying is cutting paper for the same reason.
A problem these days is that the scissors are sculpted.. they fit a right hand. But there are supposedly-left-handed scissors.. except that 99.9% of them are still crossing the blades the same way! My father bought one like that, as he was left handed. The scissors are mostly useless, at least for anything where you need that slight pressure to get a good cut.
They do in fact make a difference. Trimming my beard with beard scissors requires a bit of a strange grip in my left hand to actually cut well when I switch sides.