Medical scientists believe that the brain structure is abnormal for persons with a major mental illness such as schizophrenia. Some persons with schizophrenia, perhaps a third, have enlarged ventricles, or empty spaces within the brain, which translates into less overall ability to function, reason, and experience the environment. Other brains of persons with schizophrenia are closer to normal, with structural problems that are more subtle, and more localized to within only some areas. Because of this theory, it makes sense to believe that a person with schizophrenia will still have problems even while medicated. A lack of control of impulses is frequently one of those problems.
It is important to note that many persons with schizophrenia or other mental illnesses still exhibit brilliant levels of intelligence without being "idiot savants." What you might get is someone with a lot of intellect, but with problems in other areas. Parts of the person's brain are healthy and produce the brilliance, while other, smaller parts that may be responsible for regulating mood or maintaining mental equilibrium could be improperly developed. You then get a person whose higher functions could be intact, but who may have trouble with some basic tasks.
As a man with schizophrenia, I have experienced a deficit in impulse control, and have observed that this is not uncommon for people treated in the mental health system. It is an issue that has harmed my progress in life, and one that I continue to grapple with. Thinking through the possible results of my actions usually works to keep my behavior appropriate. If the results of a problematic behavior appear dire enough, it can help create enough "voltage" in the mind that a specific behavior won't continue.
When dealing with a person with schizophrenia, sometimes you are dealing with someone with good intentions but with behavior beyond their conscious control. A more recovered person with schizophrenia can learn to regulate their behavior most of the time.
This discussion invites the issue of competence. The idea that we have behavior beyond our conscious control implies incompetence, and this brings legal issues. Persons with mental illness should be taken on an individual basis. Some should be taken as responsible for our actions, while others are too severely ill to be considered responsible. You can't say that, across the board, persons with mental illness are either competent or incompetent. This is sort of a gray area.
The tendency to behave impulsively and furthermore to have actions that are based upon delusional thoughts can be countered and combated with deliberate training. In order to do this, you need to start with someone whose intellect is intact and functioning. This retraining may not solve one hundred percent of the problem, but it will help. By remembering that the illness is the cause of some of the behavior of some persons with schizophrenia, you can stop blaming the individual for actions that were caused by a deficit in brain functioning. It is fine to condemn the actions but don't condemn the person.