No one expected the grassroots movement triggered by Jeremy Corbyn’s candidacy, a leadership bid more about putting policies on the agenda than power. But hundreds of thousands have been galvanised, many now joining a Labour party they were previously disillusioned with. To succeed, however, this movement has to reach out. Social media have transformed political debate in ways we do not fully understand. It was once possible to go for months, even years, without a random stranger abusing you. Now it can be a daily experience if you have an opinion, on the future of British society, whatever your politics.

During the leadership race there was much scrutiny of the online abuse levelled by supporters of Corbyn, and even repeated demands that he denounce them. This was the height of disingenuousness. Corbyn and his team refused to engage in abuse while rival campaigns denounced each other as the “New Labour Taliban” or for “swallowing the Tory manifesto”. He repeatedly asked supporters to follow his example.

But the real intent of those who focused on his supporters was to link him to abusive people he didn’t even know. On a lesser scale, I’m subjected to daily social media abuse from people who happen to vote Tory or Ukip. Imagine if I demanded David Cameron or Nigel Farage dissociate themselves from this sort of behaviour.

But that does not mean there are no issues with how the left conducts itself and communicates. Online, Corbyn supporters have been compared to “cybernats” – the rumbustious and sometimes abusive supporters of Scottish independence who allegedly hunt critics in packs. And there are similarities: an unpredicted grassroots insurgency, an enthusiasm unshared by rivals, and a media and political establishment united against them.

Obviously all movements will attract a proportion who are personally poisonous by disposition, a tiny minority whom opponents will try to portray as representative. But when it feels as though everyone is against you, it becomes tempting to lash out. Even mild criticism seems to be aiding and abetting a broader conspiracy, however unintentionally. It can become difficult to distinguish between the unconvinced and the deeply hostile, the open-to-persuasion and the arch-critics.

If social media was as invaluable as the left would like it to be, Labour would now be in office with a majority of 150

Social media have proved an invaluable tool when it comes to mobilising people and challenging myths and outright lies spun by politicised mainstream media, which scarcely bother separating opinion and news.

Let’s be honest, though: if social media were as politically invaluable as the left would like, Labour would now be in office with a majority of 150. The left, and supporters of Corbyn in particular, are often accused of retreating into a echo chamber. That is an obvious danger for any individual or movement that operates almost exclusively via social media: tweet something sticking it to the Tories, start watching the retweets piling up, and it can seem as though society is cheering you on.

Corbyn supporter Owen Jones takes us behind the scenes as Jeremy Corbyn is elected Labour party leader Guardian

But if the Corbyn-led Labour party is to be successful, its supporters have to communicate. It is difficult, with relentless hostility raining down on your head, but that means adopting an inclusive, cheerful, positive approach: love-bombing opponents, even. Nearly 4 million people voted for Ukip at the last election. If they are dismissed as racists rather than working-class people who often have unanswered fears over jobs, housing, public services and the future of their children and grandchildren, they will be lost forever. Social media are important but many people don’t use them, and certainly not to debate politics. And they are not a substitute for going out and organising in communities.

What is required is persuasion, not berating. “Public opinion” is one of those misleading generalisations thrown around by the commentariat as though it is homogenous or static. Views can vary wildly according to class, age and gender. And as the refugee crisis has shown, views can suddenly shift. But that doesn’t mean being oblivious to where people are.

Most Britons want increased taxes on the rich, public ownership of rail and other utilities, and better workers’ rights; and they also want significantly lower levels of immigration and a cut in benefits for people who need them. Demonising them for holding such views will not win: only patient, clever, friendly persuasion. This means not simply relying on statistics but using stories that humanise the targets of political and media opprobrium.

Most of the British media will be institutionally hostile to anything the left does. We can complain about it, but however big our meetings and rallies we can only reach the broad mass of a public, for whom formal politics is a passing concern, through the mainstream media.

Corbyn opting to go for a community mental health event rather than appear on the Andrew Marr show on Sunday underlined his authenticity, his commitment to his constituents, his passion for political issues. Abandoning these traits would be fatal. But a media offensive with clear, sharp messages is crucial too, or a campaign of smear will be left unchecked.

I’ve sometimes been criticised for having a column in a newspaper with an editorial line that is often at variance with my beliefs, or to appearing on the likes of Sky News. But without engaging with the mainstream media it is almost impossible to get a message to the as-yet unpersuaded.

We can’t just want retweets and packed halls after all, but to change the world

And then there’s having a broad-based message, which both Corbyn’s leadership and a broader movement must have. If you’re on the left, nothing will make you feel more passionate than the plight of the poorest in society. The aim, surely, has to be to reach both low-income and middle-income people: those who work in a low-paid job and the self-employed with insecure income but a lack of rights; those trapped on social housing waiting lists and those excluded from the housing ladder; those young people saddled with debt and lacking secure work, and those older people without good-quality care in their later years.

Inclusive, outward-looking, willing to engage with the unpersuaded, hopeful, positive: that is the only way the left can thrive. We can’t just want retweets and packed halls, after all, but to change the world.