I don't think it is a problem only on the left. However, I've seen it more consistently on the left in recent times in the US. That's the only reason I mentioned it. I'm happy to hear and learn about counter examples too though.

I do not agree with the definition of 'freedom of speech' you are using, and I don't think most would. Why would it be defined only in terms of the government? It is a general concept that is useful at all times, since it is about the open exchange of ideas. An open exchange of ideas is useful in governance, at work, among friends, etc. And yes, it is useful when discussing private organizations, particularly when there they harbor immense control and face little competition to create choice (as is the case with some platforms like Twitter or with payment processors).

And as I mentioned elsewhere, Karl Popper was very clear about preferring not to suppress speech but to use rational arguments against differing opinions. In fact, using freedom of speech to suppress freedom of speech was the main problem he was talking about in the context of the paradox.