They don't really have special rights. If you're a random person standing on the sidelines watching the cops stomp some rioters, you should be safe. If the cops say "this area is closed to everyone", then you, the press, and the rioters are all required to leave.
However, if you're an innocent random bystander while all this is going on, the cops could claim you're part of the mob and sweep you up too. Ideally, that shouldn't happen, but it sometimes does when cops get angry, or overstep their authority, or the bystander starts shouting and kind-of gets involved.
The point of press markings is to call out that you are not part of the crowd, and the cops have no reason to treat you like you are. But you still have to not act as if you are part of the mob.
Quite a lot of the fake "Press" at these riots do take part, such as Arex Johnson in Portland, who does get arrested, multiple times. But it's hard for the cops to tell the difference in the dark with imperfect observation of who is doing what.
I'm all for legitimate standing-on-sidelines recorders being left alone, but I think the cops should take a harder stance on the borderline cases. They seem to be trying; when ImHiram gets up in the cop's face at the front line, they'll spray him even though he has Press markings. And they'll shove anyone who doesn't get on the sidewalk when they say to.
