It's not necessarily a bad thing to be, being insensitive, if you know how to handle it and if you can actually understand that you are. So if you are, don't worry. This can help you. Insensitivity is actually quite a good thing!
How to know if you're insensitive
Ask yourself, "do you really care"?. Okay, so someone really close to you is having a breakdown. How much do you REALLY care? Then you may well be an insensitive person.
Ask yourself whether you care more about your feelings, or other peoples. I am very emotionally selfish, and if the answer is 'I care more about myself,' then you qualify for an insensitive person.
Spend a lot of time with yourself, on your own, with no disturbances. Think back and try to recall the times when someone was in distress or if they were dismal... how did you feel within yourself? What were you thinking in regaurds to their situation?
Try writing about your feelings... it sounds cliche, but tell nobody and simply write down your wants and needs, and how you feel about particular things. Read over it and see whether your writing integrates other peoples feelings and emotions...
If you are emotionally selfish and if you are insensitive towards other peoples feelings, admit it. Don't be in denial. There's nothing wrong with it.
Don't be in denial
Be proud of it; being insensitive can help you get through in life. It helps you because it means you won't let other peoples problems get in your way. Mick Jagger is insensitive and he is so successful and he is simply brilliant.
Don't dwell on it. If you won't like this quality about yourself, it's okay! Everybody has faults.
Be happy. You may be emotionally selfish, but who cares! There are worse things in life!
Don't insult people... If they're worried or upset about something that you think is really stupid try and see things from their point of view, ESPECIALLY after you have some space to think about it. You may not be so quick to think it is so stupid next time.
If you're going to tell people you're insensitive, be careful about who you tell. Some people are really emotional and would take it personally and harshly.