I am ambivalent about it.
The trouble with it is that, if centralized, it imposes the burden of having a very large "machinery" to prioritize what needs to be responded to and maintain some degree of coherence and consistency in how all of that "incoming" is responded to. At a time when so many are kvetching about government being "too big", increasing the staff complement to deal with chitter-chatter (and all that staff will need managers and editors too) seems wasteful. Based on what I've seen happen with simply the addition of being able to e-mail the office of the head of government, there is also a very great risk of a chilling effect as the senior management of that machinery strive to make sure there is consistency in outgoing messages, and that it is appropriately adapted to all sub-communities.
At the same time, I see so many members of the public getting it wrong. And as I am fond of repeating: when transparency gets up and leaves the table, fear, loathing and paranoid conspiracy theories are more than happy to take its seat. So there is some merit in individual public servants trying to rectify misunderstanding when they spot it. Where I get cautious is when it turns into some sort of organized army, rather than individual public servants resonding to misunderstandings about their agency's actions.
It wold be nice if agencies could maintain web pages with explanations and rationales, phrased in "regular guy" language, and let their employees know what's posted on a regular basis. Then, all that would need to happen is that any employee who stumbles on a misunderstanding when they see it on some social media site or forum, can simply link to the agency message, instead of having to figure out how they're going to defend their agency's activities.
The qualifier is that whatever the agencies post has to be authentic, not bafflegab intended to deflect. It has to be sincere and convey that "This is what we are doing because we feel it is the best course of action, though not necessarily the most expedient/cheapest/pleasing/etc, for the following reasons...".