Here is an article that lists out various attempts by employers and professional regulatory bodies and their attempts to restrict the use of social media by employees and others.
Public vs. Private
One aspect of the analysis to consider is whether the entity doing the restricting is in some sense “public.” Therefore, police departments, school districts and municipalities may have a harder time restricting employee speech than do private entities such as professional football teams.
The restrictions of the Pueblo Police Department disallowing its officers to identify as working as police officers, seems overly broad. Why shouldn’t officer receive training about what is and isn’t appropriate social media conduct? They could even sign a form stating they abide by a code of conduct but to disallow them from factually disclosing their professional affiliations goes too far.