It's a bit vague.
By responsibility, do you mean why I make my choices?
Because if you were to say, "What are your responsibilities?" to me, I could say anything, like feeding my cats.
But that would be a responsibility to anyone who had faith too.
If you asked me why I had a responsibility to say, not hurt someone else, that would be different.
A person of faith might say "Because hurting a person is against the Bible/Quran/My Belief, and in doing so I disrespect God".
I would say as an atheist, my reason for that responsibility is that it's not right for me to hurt another person for a number of reasons. If they were hurt and it upset them, it would make it hard for me to be friendly with them. I wouldn't want a person to be hurt and that to upset myself.
And I wouldn't want to hurt a member of my own species, because that doesn't help my species in the long-run.
It's all a bit technical or unusual, but the question is a difficult one to answer because it's vague.
If you clarify, I could probably answer a better personal explanation.
