It's usually more of a question of breaking some other law, not livestreaming specific. Like interfering with a police officer, resisting arrest or wiretap statues. So the "mystery" isn't about the existence of a livestreaming specific statue, but if other statues apply.

Qualified immunity may be invoked if it was not "clearly established" that livestreaming was legal at the time of the incident and the police should reasonably have known about it. Going forward, it's now established in the 4th circuit that livestreaming is legal, so long as the suspect doesn't violate another law in the course of livestreaming, like interfering with a police officer.

Other jurisdictions have laws that say, to paraphrase, that it's legal to record the police making an arrest but you have to maintain a certain distance and not interfere.