Group 1: I would rather deal with harassment and hate speech than risk erroneously silencing someone I need to hear.
Group 2: I would rather risk silencing someone who's voice ought to be heard than allow someone use speech to hurt another person with harassment or hate.
---
Group 1 isn't trying to promote hate speech or harassment, they're just worried about the potentially malicious intentions of those that want to decide what speech falls under those banners.
Group 2 isn't trying to silence opinions they disagree with, they're just worried that people are getting better at subtly codifying their hate speech in a way that gives them plausible deniability to out-groups but is obviously hateful to the in-group.
I think the important takeaway is that neither 'side' is evil and despite the fact that it causes heated arguments one should maintain respect for those they disagree with.