It's a common (and somewhat right-wing) trope, but it's not really true. Thousands of people aren't arrested for "mean comments" or similar. That isn't to say the UK has no problem with the balance of freedom of expression and regulation, but it is nothing like the impression you'd get from some sources. There are some high-profile cases which are commonly misrepresented to support these claims.
It is definitely true. It's "right wing" because the soft power is with the left (or was until maybe very recently) and so only principled people of the left (e.g. Bill Maher) and all the people of the right (either principled on speech or saying things that are more likely to be censored) are going to be talking about it.
I think generally the argument is that these usually aren’t credible threats, they’re hyperbole, and the police are overstepping by arresting people for saying they’ll do something when they have no evidence that that person would actually do that. A free society doesn’t arrest someone for saying “I’m going to burn down so-and-so’s house.” It waits until you actually go to so-and-so’s house with a can of petrol, and then arrests you.
It’s the same problem as amber alerts - the stories you here about are the dramatic stranger danger ones, but the vast majority of the statistics are “boring” custody disputes.
The poster up thread said people were being arrested for "mean tweets" and now you've moved it to "but those death threats weren't serious!" No, the original claim was dishonest bullshit.