Hi Angela,
If you are acting on behalf of your client in their Twitter or Facebook account, I would ask your client whether or not they want people to know it is not them responding personally. If the question is
should you be responding on behalf of your client and not telling people, I would say that's up to you to decide what you feel is the right thing to do.
Either way, I would think that if you have a policy in place you should make it clear to your client what your firm policy is when you sign a contract.
Hope that helps!