Gethsemani got it right earlier. Police do have practices to help protect us... but we do take responsibility when we cover events like protests or riots. And something he mentioned, the part about making sure we are identifiable as press, is something I'll admit we in general don't do very well. I'm not often called out for my credentials (I'm a studio/tech guy and newsroom editor) but in situations were I have to... all I have is a laminated card and a larger laminated lanyard with ID. For the most part we are out there in street cloths as well. On the other hand, the thing about protesters or criminals claiming they are the press... that doesn't happen. Cue all the "link report here" press haters out there, but on a larger scale it just doesn't happen. Reports of actual press getting harassed because police THINK they are lying are far more common than reports of people actually pretending to be the press.
I'm not sure if it is a widespread thing or not, but working with our police department and county sheriffs we have what are called 'staging areas.' They have a watch commander tell us where to set up. If we are in a staging area, we are off limits and have every right to be where we are filming whatever. It is pretty limited however, if we aren't in the designated staging area (which easily could be too far away for good video) or are in transit to or from a staging area... we're on our own. My news director got a face full of teargas during a protest... back in the 70's (so this isn't really just a recent problem.) In that incident he was not in a staging area, he was on his way there and was crossing through the protesters. He doesn't believe the use of tear gas was justified in that instance, but he also can't really claim police were attacking the press specifically. Of the recent protests in my city everything has been largely peaceful (a store window was broken, and a police helmet was damaged) so I really don't have any recent anecdotes outside of what I read from other news sources like anyone else.
Police attitude toward us in general though... varies greatly. For the most part we work together, in an effort to keep the public safe. In instances where that is the case everything is fine. That's stuff like traffic incidents, natural disasters, standoff situations, suspicious devices. But in instances where we are covering an investigation, or something like protests... they are in full obfuscate mode. A part of me can't blame them, they can get in an awful lot of trouble for saying the wrong thing and it is our job to ask questions that they would be pretty stupid to actually answer. Basically we ask a lot of "I know you are just going to shut us down with 'the investigation is ongoing'' questions. But when in cover their own ass mode they do tend to keep us from getting information the public would like to know.
As far as our Constitutional rights to cover events. It's actually a lot like freedom of speech, we have it but there are limits. We have the right to cover whatever, but no guarantee of safety in doing so (other than from government representatives themselves.) And if their actions inadvertently hinder or harm us, that's a matter for the courts (the government court system.) They set up their staging areas, we are free to leave those areas though. They just won't guarantee our safety outside of those areas. Also; we aren't free to break the law covering a story, and they can and will charge us with obstruction if we get in their way or put them in harm's way. And for the most part government's cooperation with the press isn't compulsory. Most legitimate news sources realize we will get far more information out of the police when we cooperate with them rather than just stick a camera and microphone in their face, causing them to shut down entirely. It's a delicate balance we (and they) have to walk. We have to work with them almost every day, but we also are what holds them accountable on the occasions when it is necessary.