As someone on the far left side, most people in my political tribe have no real power (not even to really cancel people) and are constantly on the receiving end of legal censorship. For example both democrats and republicans voted to make anti-zionism hate speech recently. The left cannot do so in reverse.

Where us leftists can have problems is like you said hurting people within our own small circles. You may not be able to ruin someone's life but you can exclude them from the radical book club, the organizing committee, and all the little structures that make up your community.

To me thats more a problem of most people in our society not being able to resolve interpersonal conflicts and isn't really about censorship. Most people don't have the skills to engage in repair when someone does them wrong (or they do someone wrong) and so they just get ostracized.

Again, this is a problem that exists everywhere because we are a highly individualistic culture and as a consequence skills to keep communities together have atrophied.