Rights and responsibilities.
The phrase "with rights come responsibilities" is one of those magical phrases that cleverly hides the nature of what is being discussed. The phrase is most often used by someone in a position of power to get you to make concessions in some area other than the current one.
For example; yes it is your right to have some maternity leave, but you have a responsibility to the company and your fellow co-workers not to place an undue burden on them.
A true right is not negotiable, otherwise it becomes merely a privilege. If you have to satisfy any conditions in order to qualify for a right it becomes merely a reward. Taking the rights of people and turning them into currency in order to exploit them is usually what speakers of this phrase are attempting to achieve.
The basic human rights, of life, freedom, and equality should never be dependent on good behavior. That is not to say that rights cannot be denied, it is possible to kill, imprison and subjugate, but doing so must be as much of an anathema as our support of such rights is of primary importance.
For one person to act irresponsibly can not negate another's rights, it can however call for justice.
If we have any responsibility it is first to ensure that rights are maintained.
Of course we all have responsibilities as well, but these are mostly negotiated culturally and in context to our situation. Rights and responsibilities are very strange bedfellows, be cautious when you hear them mentioned in the same breath.