I think you assume responsibility when you think someone who is on-duty does not behave in responsible way so you do the job without authorization.
Are they used also in the following senses?
1.The state or fact of having a duty to deal with something or of having control over someone:
Women bear children and take responsibility for childcare
2.The state or fact of being accountable or to blame for something:
The group has claimed responsibility for a string of murders