Occasionally, I see some commenters argue that attempts to restrict speech are chiefly a trait of the left (even defined broadly not just to include the hard Communist-or-near left) rather than of the right. Here's an example, from a recent thread:
I don’t think that’s true that the right is “just as guilty” as the left when it comes to censorship. Most of the examples of government attempts to suppress political speech seem to have come from the political Left. Wilson and Roosevelt were particularly notorious for it during WWI and II. We had Al and Tipper Gore’s attacks on the music industry during my childhood but I honestly cannot recall any similar or comparable attempts at censorship by the political right.
This seems to me to be a mistake. My sense is that at least from the 1950s to somewhere around the 1980s, the left has generally (not entirely, but generally) been more speech-protective than the right; since the 1980s, matters seem to have roughly equalized (recognizing that it's impossible to measure things precisely), but there still remain plenty of calls for restrictions from the right.
Just to give some examples, the defenders of Communists' and near-Communists' speech in the 1950s were, to my knowledge, mostly associated with the left (and not just the Communist left); the restricters were both on the left and the right, but the defenders were mostly on the left.
The defenders of protection for sexually related speech, and not just pornography but serious literature and even political advocacy, have been mostly on the left, and critics of such protection have been mostly on the right. (See, e.g., Sex and God in American Politics: What Conservatives Really Think, Pol'y Rev., Summer 1984, at 12, 24 ("I don’t think the advocacy of homosexuality really falls under the First Amendment any more than the advocacy or publication of pornography does." (quoting Irving Kristol)). The advent of the left-wing feminist calls for restricting sexually explicit speech in the 1980s has evened the matter somewhat, but I'd say that on balance the right to engage in sexually themed speech still gets more support from the left than from the right.
From the 1980s until now, campus restrictions on allegedly bigoted speech have been largely the province of the left (though many on the left have opposed them). Conservatives have generally insisted that even epithets ought to remain protected, at least outside the rubric of fighting words. But in the 1970s, many conservatives took the view that profanities and epithets should not be protected, including in university newspapers, see, e.g., Papish v. Board of Curators of University of Missouri, 410 U.S. 667 (1973). My sense, though I am not an expert on this, is that in the 1960s and the 1970s many conservatives were willing to see a fairly broad range of restrictions on what was seen as offensive or radical campus speech, while liberals generally supported protecting it.
Likewise, the movement to ban flagburning has been mostly a conservative movement. The (fortunately quite rare) calls to restrict anti-war speech have been chiefly on the conservative side. Calls to restrict violent rap lyrics that seem to urge attacks on police officers have often (my sense, generally) come from the right, see, e.g., Chuck Philips, North Steamed at Ice T; He Wants Time Warner to Face Sedition Charges Over Rap Song, L.A., July 2, 1992 (describing Lt. Col. Oliver North's suggestions to this effect), though calls to restrict violent music and videogames more broadly seem to have been pretty evenly balanced.
There are some areas, of course, where calls for speech restrictions come mostly from the left and calls for speech protection mostly from the right. The resistance on campaign finance speech restrictions has mostly -- though not entirely -- come from the right. Similarly, at least on the Court, as to the resistance to speech restrictions on judicial candidates, and the resistance to the Fairness Doctrine (which restricts some speech to advance other speech, and at least sometimes is urged by some liberals precisely because they dislike the current tenor of broadcast speech). The resistance to broad and vague "hostile environment" speech restrictions has generally come from the right, though some on the left have joined it, and many on the right have sat it out. And restrictions on anti-abortion speech, including restrictions that go beyond simply banning the blocking of abortion clinics, have generally come from the left.
Nonetheless, the big picture is both the left and the right calling for some speech restrictions, and opposing other speech restrictions. One may certainly support the right's preferred speech restrictions and oppose the left's, support the left's and oppose the right's, support both, support neither, or support some from each side (most of us would endorse some speech restrictions, even as we would oppose others). But there's little basis, I think, to claim that "Most of the examples of government attempts to suppress political speech seem to have come from the political Left."