Is he or she the face of the organization?
Is he or she the voice of the organization?
What is their role? To tweet, or to lead?
The leader should focus on creating a culture of communication.
If they have time after that, and it's appropriate to their role, then It's up to them to be participating in social media on behalf of the cause, and only if it's a good use of their time.
It could be a waste of time and a distraction. It is not as important as other things leaders should focus on.
Leaders should focus on what matters in their role. It’s not always the leader’s responsibility to be the CTO (Chief Tweeting Officer).
Lead first. Tweet second.