This is going to sound like a blow off, but it depends on the scar. I’ve got two scars on my lower lip, that I got within a couple weeks of each other. The first one was a blow from a staff to the face, it tore open my lower lip, and actually knocked two of my teeth out of position, where they remain twenty years later. There’s still a visible scar in the lip, but there’s no deeper sensation with the injury.
The other one was an elbow strike to the opposite side of the face. Honestly, I can’t even find that scar. But, if I twist my mouth in just the right way, it feels like my lip has been painlessly ripped open… It’s there, and under just the right circumstances I can feel it.
So, it will depend on the specific scar, but it can happen. I’m not sure about larger scars on the side of the face, but it wouldn’t surprise me.