Religious conservatives often contrast "responsibilities" with what is seen as a progressive (over) emphasis on rights, at the exclusion of any discussion of responsibilities. Very often, the take-away for conservatives is wanting to talk more about responsibilities - or at least both rights and responsibilities equally. Responsibilities are typically seen as the duties and obligations people have towards each other, to their country and to God in terms of how to act and behave.
Liberals often hear in the word “responsibility” a coded message that discounts structural/environment/societal causes of social problems (racism, oppression, discrimination eg) and justifies reducing or eliminating government programs to increase opportunity/economic security (e.g. welfare).