'carry a responsibility' is correct and usable in written English.
You can use it in a sentence to express the idea of someone having to do some task or to take on some duty. For example, "As president, he had to carry a responsibility to make difficult decisions for the good of the country."
That is all very well, but we also carry a responsibility for what we say.
I have to admit I found his argument slightly odd, though I agree we all carry a responsibility to share accurate information.
With situations such as the high-profile technology breakdown suffered by NatWest earlier this year, jobs like Anthony's carry a responsibility that has both advantages and disadvantages.
Players are aware that their decisions carry a responsibility that ours do not.
We carry a responsibility now to look after that and to prepare ourselves, control what we're doing this year and play some very good cricket.
"I challenge you to raise issues regarding child rights: the right to quality education, right to participation, right to protection against violence" - UNICEF Representative in Azerbaijan, Hanaa Singer "Your presence here today is an honour to you, but also you carry a responsibility to make sure that the voices of the children who elected you are heard.
Whoever rules this vast country, with its enormous economic strength, its military potential, its cultural influence, carries a responsibility which is felt all over the world.
But let's not forget: Freedom also carries a responsibility.
"Those of us on the left and centre-left carry a huge responsibility.
"They may look extreme, but now they carry a huge responsibility to deliver results on the ground.
Vince Cable, the hammer of the banks which carry a heavy responsibility for the deficit the country faces, is not to be at the Treasury.
