As to what the state's responsibilities are surrounding freedom of speech
technically: I suspect the courts will figure that one out.
As to the intrinsic responsibilities of editorial control: Sure: legal obligations are what you define them to be. But escaping legal consequences doesn't mean you're not conceptually responsible; it simply means the law isn't perfect. And of course there are lots of ways you can choose to distribute responsibilities. So perhaps I should of said: I don't see a way for the considerable influence that media platforms have to be held accountable except if they are also responsible. The obvious alternatives are that that power is completely unchecked, which is harmful, or that somebody else dictates what they can do - which I kind of doubt will work in the first place, and even where it does, that means you better have a lot of faith in those controllers. Simpler and more reasonable for the media platforms to actually be responsible for what they say, and not maintain the fiction that they have no influence or effect on the messages they "transmit".
As to requiring a judge to get involved before censorship: I think that's actually worse than complete immunity. Involving a judge makes it appear as if there is control, and as if media platforms are held to some kind of unbiased or at least fair standard. But the whole point of a modern media platform is that you don't need manual control; and indeeed you can never keep up with manual control. Algorithms will win against judges every single time, simply because no society can have that many judges (not to mention even judges aren't a panacea: fairness isn't trivial). Judicial interventions will necessarily be only in such tiny fractions as to be meaningless - and even if punishment and enforcement were scaled to absurd levels (with their own costs), well, media platforms are international behemoths. A judge may well rule some post to be illegal, but many posters will be beyond the judges reach.
If media platforms get to exercise editorial control of sorts without being held responsible, then they will influence the debate in ways that undermine the very reason for freedom of speech to exist in the first place. I kind of expect that outcome is what will actually occur, and the consequences in the long run are hard to predict, but I'm guessing: dire.
The incentives for a media platform are all wrong. They want to push messages that encourage using the platform; and as such they're going to be enablers for needless tribalism, short, poorly-thought-out posts, for anybody who is willing to pay to influence regardless of the merits, for trying to shield users from being confronted with evidence that they're wrong, and more.
If you will: I think people have started to accept that the idea that metadata is somehow something completely different from data is false. Metadata is data, and similarly; the meta-control a media platform has over messaging is really huge. If anything: the fact that they don't actively edit the contents of the messages only grants them more power, because it makes the control they do have opaque and hard to judge the impact of. I don't mean to imply they're intentionally Machiavellian or whatever here: it's just that indirect, hard-to-observe control is really hard to value.
----
So, to go off on a tangent....
I sometimes wonder what the historical figures involved in the evolution of modern freedom of speech would think if they saw what actually happened. For one; we've grown to utterly fetishize the concept to the extent that rational thought has little to do with it. But simultaneously, it's like we're walking around with blinders on: we don't have freedom of speech, and never will be able to get it absolutely, and the idea that that would work or was the idea some long time past is... well, not exactly founded in anything.
And then, of course, there's the fact that obviously times change, technology changes, society changes, and it's pretty crazy to think that concepts like freedom of speech are somehow impervious to all of that. What worked hundreds of years ago may actually need some tuning. Yet that idea isn't exactly well received in my experience, at least not in the US: thou shalt not question The Founders.
Ironically, there have been some pretty major changes to how we use freedom of speech, and in ways that makes me feel like it's gotten worse. As if we're playing telephone/chinese whispers, and people are going through the motions even if the motivation to do so would seem obviously lacking to those in the beginning of the chain.
There have been obvious changes such as copyright evolution; but also other changes; e.g. lobbying was for a time seen as fraud; not protected speech.
So on the one hand we cry about the inviolability of free speech; yet on the other we've changed what that means repeatedly, and the world has changed too. I don't think that's a healthy mix.
Social media (and more broadly the data-science era) are still young. So far, the way they have affected public debate is probably still relatively random. There may have been been damage, but it's probably rarely been malicious. So whether you support the status quo - social media free from control and any obligations to the public - or a more constrained media somewhat depends on whether you think that's going to stay that way.
I'm not betting on it.