You are using empathy in a different sense than (I think) most people are in this thread. I'm no psychiatrist, but usually when I hear people discuss empathy (or more often, someone's lack of it), they mean it almost in the sense of a lower-level mental process. Empathy is more than just feeling bad for random strangers who feel bad themselves- if anything, that's just a side effect. Empathy is the thing that makes you feel bad when you hurt someone else, even if you had a good reason for hurting that person. Empathy is the thing that allows you to sense other people's internal states- it's not magic or psionics or whatever, it's just your subconscious maintaining a background thread that is paying attention to signals from other people, and communicating those signals to you by internalizing a distant echo of what that background thread has determined that they must be feeling. It's fundamental to human interaction and cooperation and is (IMO) one the biggest drivers of the development of civilization.

A sociopath is an adult who never developed this facility. Children don't have empathy- it's one of the things that makes them children. That's why some kids do things like torture or kill animals...in their minds, those animals aren't creatures with sensory experiences like themselves, rather those animals (or even other people) are perceived as meat robots, for lack of a better term. Lacking empathy means when they hurt someone, they don't feel anything themselves because there is no background thread in their subconscious reminding them that the person they are hurting is another thinking, perceiving, feeling being like themselves.

Psychiatrists can't diagnose someone as a sociopath until they're 18, by which time people's brains are expected to have developed some capacity for it. People on the autistic spectrum also have a lesser capacity for empathy, that's part of what defines them as being autistic. They can't perceive subtle social queues because of this lack of empathy.

There are plenty of people who are self-aware, functioning sociopaths- they've spent their lives having fundamental social problems, never relating to other people the way that everyone else seems to be able relate to each other. At some point in life they discover that their cognitive development wasn't perfect, and once they understand the nature of the issue they can compensate by making a conscious effort to do so. There are also plenty of sociopaths who never made this leap, and basically all you can do is stay as far away from them as possible, because who wants to deal with someone who thinks you're a meat robot?

In the context of this thread, people aren't talking about sociopaths or anyone with autism/Asperger's/etc., but imagine a seemingly normal adult with underdeveloped empathy- it manifests in all sorts of ways, from being jerks to retail employees to stiffing waiters to just being an unpleasant person to interact with. Those people don't realize they have an underdeveloped cognitive process, they just go through life wondering why so many people think they're jerks, and saying things like "99% of humanity is way too sensitive". I would attribute a lack of self-awareness to a lack of empathy as well- if someone is incapable of seeing themselves through other people's eyes, then they have no way of gauging their own behavior or feelings.