Government Agencies should not restrict their social media efforts to only their own social media channel/sites. They should each have a person or a team of people who search for their topic(s) on other social media channels/sites, blogs, newscasts, etc. They may find incorrect or incomplete information or misconceptions. They may find that they have something to contribute to the discussion, something the others had not thought of. For example, I have done some looking in respect to my own website and I have found a number of blogs that reference incorrect/broken URLs. In each case, I was able to identify myself and provide a comment with the correct URLs for the information they were discussing. I was also able to add a little to each discussion while I was there.
The whole idea of social media is to go where the customer is and partake in the conversation. If you stay merely to your own social media channel/site, then you are not really going to where the customer is and you are missing out on the discussion.