Short answer is no, but a level of awareness for how you're being perceived is an asset.
Generally, I care what other people think of me. I want to be liked, which means that if somebody doesn't like me, I might've done something wrong in their eyes and so it bothers me. But, as we all know, we have no control over what people think of us. Not everyone will like us, not ever. So it's wasted energy to care too much what people think.
But seeing as we live in a society, empathy and a general awareness for how people might react to your actions is important. So while you shouldn't care what people think of you, you should act in a respectful manner.
I tend to care a lot more what my friends & family think of me than work acquaintances. Work people's perceptions only matter in terms of work ethic and ability, in my experience. Friends and family are the relationships I want to foster.
Also, closely related is embarrassment, and how people see you after you've done something embarrassing. It can be very hard to handle, but generally speaking, people are mostly thinking about themselves, and the memory of the embarrassing thing you did will fade much quicker in their minds than your own. Whatever was done, it's in the past, and if somebody insists on bringing it up, cut ties. They're not good people.