What people typically refer to is Antisocial Personality Disorder, which varies wildly. Many people with APD aren't criminals, or even "bad" people.
It's rare to the point that I suspect that you could house the number of violent criminals with untreatable Antisocial Personality Disorder in a single prison.
California has at least 4 state mental institutions devoted solely to housing violent mentally disordered offenders.
I'm talking specifically about people you'd qualify as psychopaths/sociopaths as they're colloquially depicted. The "evil" emotionless/unreachable criminal is incredibly rare.
>California has at least 4 state mental institutions devoted solely to housing violent mentally disordered offenders.
This is unrelated, but keep in mind that often these institutions are treating a symptom and ignoring the disease. California probably wouldn't need 4 state institutions if they had reasonable access to mental health and stable housing.
They often have zero support system when they're released. They're temporarily put into group homes or halfway houses where they're surrounded by other people with mental disorders and various related issues like substance abuse (not to mention that abuse in general is rampant in this type of housing, including by staff). Some people know that this type of housing is problematic, and opt for homelessness instead.
They've got no permanent homes, finding work is incredibly difficult, and they're grouped together with other people with wide-ranging mental disorders and other problematic histories.
It's hard to imagine that anyone is _ever_ able to escape these cycles of institutionalization.
Moreover, people with antisocial personality disorder aren't emotionless. They're non-empathetic, meaning they have no regard for the emotions of others. Most people with APD are fully capable of feeling and expressing their own emotions.
If you want to re-define these terms to mean something other than what they actually mean, you could say that the evil emotionless, unreachable criminal is incredibly rare. That's because evil people, emotionless people, and unreachable people are all incredibly rare, so the combination of all 3 is even rarer.
Criminals with APD and Psychopathy as depicted in popular media are incredibly rare — that's the entirety of what I'm trying to express. The American obsession with prisons has next to nothing to do with the prevalence of people who fit the layman's understanding of "sociopaths" and "psychopaths." If it were directly related very few prisons would need to exist at all.