1) wildly overpowered in terms of speech
2) frequently accused of censorship
3) pretty damn globally influential
I would love for the US not to be so influential in this regard, but that ain't the world we're living in.
I'm curious to know what the right effectively suppressed in 1980-2010, which is the time frame you mention. Those were pretty free decades.
I do not recall censorship. Also, the current "left" is not really left outside of cultural wedge issues. They are elitists and academics with useless degrees protecting their income streams and they despise the actual working class.
So "complete" free speech is a resoundingly absurd idea, in no way were they protected and able to speak freely. As I mentioned above, the suppression (true, I wouldn't say censorship) resulted in a sort of de-facto very unfree speech.
There are a couple of commenters here who live in India who would disagree with this idea. Here's one: