How do you solve a problem like Vladimir? Opinions about the solution to Russia's President have proliferated over the weekend. The two-year prison sentence handed down to Pussy Riot for "hooliganism motivated by religious hatred" has sparked an international storm of punditry. Outrage has been expressed. International musicians including Madonna, Paul McCartney and the Red Hot Chili Peppers have spoken out. Protesters have worn Day-Glo balaclavas. Everyone agrees: something must be done.

But to pin the fate of Pussy Riot on to one man, as though Putin runs Russia single-handedly, is misleading. He runs a powerful machine, certainly, but there are millions of active cogs inside the Russian regime, and there are many other passive participants who are allowing this to happen. Once the silly season is over, the world will once again stand back as the state machine continues its relentless project to dismantle Russian democracy and civil liberties.

Who's standing back, you say? We've sent literally loads of tweets about it. Some of us have even been to the Russian embassy to protest. How many of you? Oh, at least a hundred. Well congratulations to those who stood up to be counted, but where was everyone else?

Back in 1948, when Eleanor Roosevelt presented the Universal Declaration of Human Rights to the world, she made clear that "every individual and organ of society" had a role to play in the advancement of human rights. Human rights don't belong to states; they belong to people. And it's our job to fight for them.

Every major achievement in the history of human rights has been the result of mass movements of people – the collapse of Apartheid in South Africa; the fall of totalitarian regimes in the Soviet Union and Eastern Europe; the end of segregation in the American South. By coming together across political divides, people can do more than any government. Our rulers are limited to technical solutions – new laws, new policies – but mass movements of people can shift cultural norms, so that it becomes unacceptable to imprison people for expressing their political or religious opinions.

Yet we personalise the complex predicament facing Russia into one man, Putin, as though it's a simple story of good versus evil. We watch developments like the sentencing of Pussy Riot as though they were plot twists in a mini-series. What will happen next? Can Paul McCartney persuade the President, in this hour of darkness, to let it be? They did meet for tea in 2003 but is it fair to expect Sir Paul to carry the world on his shoulders? Check for updates.

Twitter and other social-networking platforms have given people the impression that fighting for human rights is easy: all you have to do is hit "retweet", and the world will immediately become a better place. Two things are missing from this equation: money and a movement.

The Russian government has learned from the experience of previous autocracies, and is attempting to silence Russian NGOs, not least by preventing them from receiving foreign funding. A new law requires any NGO that is funded externally to register as a "foreign agent". This has slowed down the flow of resources to such groups – but they were not receiving vast amounts from the West in any case. Between 2009 and 2012, according to research by international human-rights funders, only $15.7m (£10m) of non-governmental foreign funding for human rights went towards groups in Russia. Of the other Brics nations, only Chinese NGOs, which face similar restrictions, received less ($15.4m), while human-rights groups in England received a total of $125.9m in the same period.

Of course, some of this funding allows human-rights groups based in the UK, such as Amnesty, PEN and Index on Censorship, to campaign for free expression in Russia, but a movement depends on strong partners inside and outside the country. International donors and activists urgently need to find ways to rebuild the human-rights movement. The capacity for campaigners around the world to connect with each other, and with the general public, has never been greater. We are used to petitions with millions of signatures. The first billion-signature petition may only be around the corner. And yet something is missing – the genuine human connection that was critical to previous global movements. People can only take risks when they know and trust each other as people, not just avatars. And profound social change does not come without considerable risk.

Pussy Riot may not be the world's greatest band, but the trio of young women who have faced up to their ordeal with grace, courage and good humour are as inspirational as any Olympic athlete. Their imprisonment could be the catalyst that helps to rebuild the international human-rights movement. Perhaps this will bring about genuine change in Russia and elsewhere.

One sign of hope is the (belated) support of Madonna for the group. No one is more market-sensitive than Madonna. If she believes that there is a mass audience for human rights, then we may be on the cusp of a new era of global activism. I hope she's right.

Jonathan Heawood is director of programmes at the Sigrid Rausing Trust, a charitable foundation promoting international human rights