Like many here I would also describe myself as being rather empathic, rather than having none at all. My psychologist has actually explained that it is a common misconception that aspies or others on the autism spectrum don't have empathy - the truth is, they express it differently, and in my own experience at least, more honestly than NTs.
My desire to sort out people's problems has lead me to study psychology myself and some day I hope to work as a clinical psychologist. Sometimes other peoples' problems can be very draining indeed, but what I have learned to do is associate with a small group of friends (perhaps around 8-12 people) and I will usually see at least one of these people every second day. Many of them do suffer from depression and other disorders, and the understanding I have with these friends is that they need to be sensitive about how much they share (and also HOW they share) their feelings and their problems. I tend only to hang around people that are sensitive in this way. In return, when I am feeling depressed or down, I do the same for them.
It also helps to learn detachment (which is VERY much the opposite of not caring). Detachment is the ability to view oneself as an outsider for the sake of problem-solving. All psychologists, NT or autistic will need to learn this, or they will likely commit suicide 5 years into the job. I wouldn't know exactly how to teach detachment, but since I'm aiming this mostly at aspies, I would say the following: Think about your emotions and feelings as related to others LOGICALLY. Reason in your own mind that if you are detached yet you care, you will be more efficient at solving the problem than if you dwell on the pain that you feel.
If you do it enough, it basically becomes 2nd nature, just as another poster mentioned can happen for empathy itself.
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Aspie score 156/200
NT score 56/200
Diagnosed at 19