You'd run into quite a bit of trouble with the first amendment and freedom of the press.
While freedom of the press in the US may be violated pragmatically, you'd have a much harder time challenging it philosophically. That is, police offers arresting individual reporters (especially ones involved in protest) slips by relatively easily, but a federal law placing limits on what information the press can publish would receive strong opposition.
Are you sure about that? If a government passes a law restricting speech because it has to, and then enforces it to restrict speech, it isn't clear to me that's worse than a government which doesn't bother with laws in the first place. Either way, you have no freedom of speech, but at least with the former you still have some semblance of rule of law. That's more important than speech, IMO (although of course without both you're already pretty fucked).
But this is one situation that DID exist in 1787. Surely there were criminals in big(ish) cities, and surely the press was reporting on those criminals. Perhaps social media makes press errors more damaging, but the principle—protecting the privacy of criminals and victims—very much existed back then.
There is an established legal basis for allowing publication of victim's names: "Thus, it is ordinarily unconstitutional for a state to proscribe a newspaper from publishing the name of a rape victim, lawfully obtained. This is because there ordinarily is no compelling governmental interest in protecting a rape victim’s privacy" (4). IANAL, but permission to print criminal's names (which are obtained legally) follows from similar logic.
[0]: Congressional Research Service: Exceptions to the First Amendment: http://www.fas.org/sgp/crs/misc/95-815.pdf