The way I see it is that the European style gives the government a monopoly on recording in public.

I'm not really sure how that follows, they too are bound by the laws protecting the privacy of those being filmed. For instance license plate readers are not legal and had to be taken back down in places where regional governments put them up. I'm quite glad of this recently as more and more bad news regarding Flock and Palantir comes out.

How does that work in Europe? You see someone committing a crime and you just keep your phone in your pocket? Because it’s against the law? “Excuse me, sir… May I have your permission to record you beating this person to death? Thanks!”

The weighing of legal interests applies in such cases. The violent person may have a legal interest in their informational self determination, but the victim has a legal interest in being aided against the violence, even if it's only by the indirect measure of creating proof, and the public also has a legal interest in the prosecution of crime. The latter two are weighed more highly that the former.

It's the same fundamental idea as the smashing of other's car windows being illegal, but becoming legal if it is committed during aiding people in mortal danger.

Regarding police: Courts have determined that the public has a superior right in keeping police in check, compared to the right of privacy of the involved officers. And thus filming them during operations is allowed. On the other hand just filming them when nothing is happening, or if they are eating their bread rolls on break is not allowed, because then their privacy is the only issue at play.

It makes things a little more difficult to determine, I admit. But then I think we see police pushing their boundaries either way. We had a case go to the Federal Court (of Switzerland, we're also a Federation) recently where the police were taken down a peg, after one of them blocked a camera claiming the bystander had to follow his order to stop filming even though she was not interfering with their operation and simply documenting.