I am pretty much in agreement with what ReubenJ and others have said. Those atheists who say this, and I have known more than a few over the years, really are not talking about “personal responsibility” but about the source of moral behavior.
A religious adherent naturally derives his moral perspective from the morals taught by his religion. An atheist who believes religious people are free of personal responsibility, means that he decides, picks out for himself, what is right and wrong. So he takes “personal” responsibility in the sense that he does not rely on other, higher moral/spiritual authority.
What that atheist fails to understand is that a person takes, or at least can take, as much personal responsibility in deciding to follow a religious path, to adopt a religion’s moral teachings, and to behave according to a religion’s precepts, as an atheist can take “personal responsibility” in rejecting religion, moral teachings, and acceptable social behavior. The real difference that is being made is whether one is open to the insights and experiences of others, or one wants to start from scratch and “figure everything out for himself.” But even that latter sort of “personal responsibility” is dependent on others. They may not call it religion, but it has religious, moral, faith content. It cannot be otherwise.