Short answer, yes.

Short answer, yes. Long answer, click the link. And remember, this doesn't apply to only government agencies - it applies to corporate agencies as well because, believe it or not, your employees are social media'ing and in many cases they're talking about their work. Or, at the very least, they're connected to their peers and many posts made online can affect the relationship in person.
http://www.govtech.com/social/GT-September-2017-Should-Your-Agency-Do-Social-Media-Training.html