My experience is mixed.
I have been told that I'm a very caring, generous and easy to talk to individual. People recognise that I care about them very much - even if I don't know them particularly well. I do want the best for everyone and I do all that I can to help people.
That said, though nobody's said it outright, I do know that I probably come across as insensitive. In fact, I see it as practicality, but sometimes people don't want or need practical solutions - they just want some sympathy and for someone to validate them, and I'm not as good at that as perhaps I should be. For example, if someone tells me how much they hate their job, my response is more likely to be "apply for new ones, then", than a simple "aww, I'm sorry to hear that". I do the whole "aww" thing, but quickly move on to solutions. And if people complain about the same thing for weeks/months but make no effort to change anything, then I'm not going to keep going with the sympathy.
So yes, I probably do come across as a bit insensitive sometimes, but it's not done out of malice but out of trying to help people. And, generally, people do recognise that.