Within a few hours of my meeting with playwright James Graham we are following each other on Twitter. This feels appropriate and cosy in 2014: the right level of intimacy for two people who don't exactly know each other but have a connection. Graham is a fan of Twitter. He has been on it since the 2010 general election. He admits he has "calmed down" in recent years, and no longer feels compelled to tweet every detail of his existence to unknown others (such as "sunsets or dolphins").

Graham describes himself on his Twitter bio as a writer of "plays and emails". This understates his success as one of Britain's most interesting and original young playwrights. This House, his witty political drama set in the whips' office of 1970s Westminster, transferred from the National's Cottesloe theatre to the Olivier, following critical acclaim. In the autumn of 2012, he and Josie Rourke, the new artistic director of the Donmar Warehouse, began discussing how the digital age had transformed human behaviour. Was there, they wondered, a way of turning the modern obsession with social media – the minute self-chronicling of everyday lives, to the point where many people even photograph their food – into a theatrical experience?

What followed was a "quite gentle investigative process". The two spent a week brainstorming in Rourke's London kitchen. She considered commissioning a work on the Leveson inquiry, only for the National Theatre of Scotland to beat her to it. "I got interested in the technology around Leveson. What is phone hacking? What are the legalities?" Rourke says. Graham, meanwhile, was busy with This House.

Then, last summer, a 29-year-old contractor working for America's top-secret National Security Agency gave the project an extraordinary dramatic hook. Over the past 10 months, Edward Snowden's revelations – based on documents he swiped and leaked to Guardian journalists – have transformed the way we think about the issue of privacy. The man himself is holed up in Moscow. What once seemed fantastical or unreasonably paranoid or bonkers is now leaked fact.

Thanks to Snowden we know the US and UK governments routinely spy on us, hoover up our private data and store it. This omniscient surveillance captures most of what we do online: web searches, email addresses, headers. The intelligence agencies can do amazing things. They can geo-track our movements. They can help themselves to our selfies. From all this you can tell a rich electronic story of someone's life: joys, sorrows, loves, secrets.

Snowden doesn't appear as a character in Graham's new play Privacy, which opened this week at the Donmar. Instead, he is a giant off-stage presence. Graham likens Snowden's dramatic role in Privacy to that of Margaret Thatcher and James Callaghan in This House: neither of them actually come on stage; they inform the mood from outside. "I felt the British media focused too much on Snowden, and on his personality and motivations. It's the content he revealed that really fascinates me," Graham says.

He has written a vibrant and forensic piece of documentary theatre. Privacy is based on extensive interviews with about 60 people from what might be called the digital frontline: MPs, journalists, teachers, lawyers, hackers and activists. There are conversations with psychoanalysts, techies and teenagers (the digital forest's quintessential natives). In the show, parts of these conversations are performed verbatim on stage by actors, in the manner of David Hare.

There are dramatic re-enactments too, as well as a slightly strange element of interaction with the Donmar's 270-strong audience. (Graham doesn't want to reveal too much at this point, though he does hint at a Brecht-meets-the-iPad moment. One suspects theatregoers will be encouraged to keep their electronic devices on, with live data used to demonstrate some of the content of the Snowden files. The idea? To show how our gadgets may leave us vulnerable.)

Six actors play a multitude of parts. They take on 30 characters. These include Paddy Ashdown and David Davis (a leading critic of creeping surveillance), as well as the former director of GCHQ, David Omand, and a shadowy figure from government, responsible for national security, whose identity isn't revealed. There are judges, police, Google and Facebook. And Clive Humby, the mathematician who invented the Tesco Clubcard. Humby's shopping algorithm reveals pretty much everything about us.

Malcolm Rifkind – the former Conservative foreign secretary who chairs parliament's intelligence and security committee – features too. Since Snowden's revelations first broke in the Guardian last June, Rifkind has vigorously defended the security services. "He was brilliantly eloquent about how he thought oversight actually worked in this country," Graham says.

Amusingly, the same actor who impersonates Rifkind also voices Alan Rusbridger, editor-in-chief of the Guardian, who was forced by the government last summer to destroy the Snowden hard drives in an underground car park. (This happened after David Cameron sent his emissary, cabinet secretary Jeremy Heywood, to bully the paper.) The actors don't try to do a literal impression of Rifkind et al. But they have listened to the original audio files, and capture their subjects' vocal and physical attributes in an enjoyable way.

Graham appears in Privacy as "The Writer", with Rourke featuring as "The Director". "It's gently satirical," she says of their on-stage avatars. Graham adds that while writing his new work he was encouraged to examine how much of his own data was out there in cyberspace, a process of discovery that left him "shocked". Julian Assange, meanwhile, is missing from the action. "We made a conscious decision not to include him. I felt a lot of that story has been explored," the playwright says.

So are we, I wonder, living in a moment of techno-heaven or techno-hell? Aged 31, Graham is more enthusiast than doom-monger for our digital age and its endless micro-blossoming. "I do think there is a huge joy to now," he says, sharing a bottle of wine with Rourke in the Donmar's Covent Garden rehearsal space, at the end of another long day. He adds: "We didn't want to ape the tone of a lot of the stuff we would read or hear pre-Snowden. Theatre's particular response [to the question of privacy] has often been dystopian and apocalyptic."

Graham says he still gets a "huge amount" from Twitter and Facebook. "I get to see how happy and child-birthing and married my friends are, which I'm not." Really? "There is nothing that makes me sad that everyone is having a baby," he insists. He says he now only fires off a couple of tweets a week. "I used to feel [there was] almost a quota, and think: 'I haven't tweeted for four hours.' Cutting yourself off from the internet – when on holiday, for example – can engender "a lifting of your brain", he observes.

Graham makes a point about how there is "massively" less privacy than five or 10 years ago. "The pressure to continually narrate our experiences to a bigger, wider group is really high. I do think we are less private, politically, socially, emotionally, culturally. But I don't think we've got to the point where we can't row back from that." We live with the reality that "you treat yourself like a brand", which wasn't true for his parents, he says.

He hopes his new play has a political relevance or "urgency", in the same way that This House – set 40 years ago – addresses the very topical problem of coalition politics. "People have been dealing with issues of privacy, in art and everyday life, for more than 4,000 years. But I think something has accelerated in the last 10 or 11." The human need to share information is in-built, he thinks, but says: "These platforms allow us to do this to an extent, and to an audience that is far greater than for any other generation that has ever lived on the planet."

Rourke comes to Privacy fresh from the sellout success of the Donmar's Coriolanus, starring Tom Hiddleston. She says she has always been interested in technology and describes herself as a "bit of geek". "I've probably got more apps than you," she tells me.

What interests Rourke about the digital age is how it has affected the universal themes that theatre seeks to illuminate – boundaries, conflict, "how romance occurs, how sex happens". "There is a reshaping of our ideas of ourselves occurring through the way we interact online," she says. "In our country and the west, human beings are shifting in the way they encounter each other via these devices. Something a bit tectonic is happening."

At 37, Rourke is old enough to remember life as a young adult before the internet existed. She recalls how she and her mother wrote letters to each other while she was at Cambridge reading English, the first person from her family to go away to university. They kept in touch as mother and daughter might have done centuries earlier: "They [the letters] were like Victorian correspondence. They would say things like: 'I'm going to go now and put this in the post.'"

Does Rourke feel nostalgic for this calmer, slower age? Should we think of it in prelapsarian terms? Yes, she says, but: "It's easy to become overly romantic and a tiny bit luddite about it." Both writer and director think they have one foot in the analogue world and one in the digital. Graham chips in: "I remember the world pre-mobile phone and internet. My God, it was a simpler time." ( Tim Berners-Lee, a prominent Snowden supporter, came up with the idea for the world wide web back in 1989, when Graham was only seven, growing up in the mining town of Mansfield.)

Rourke talks fluently about big data, the algorithim-driven omniscience of large corporations, and how her dad and brother still don't understand what metadata is. She is fascinating on how technology has changed the rhythms of everyday life: "In some sense, the mobile telephone has replaced the cigarette in public. It's a thing you can do on your own, in longueurs, nervously. When you are on your own at a party and feeling a little bit shy, instead of lighting up you take out your phone."

It is a moot point whether Cameron – and other members of his gilded Oxford generation – would have been able to enter politics if iPhones had been around in the late 1980s. There isn't much of a visual record from back then, with Cameron seemingly keen to extinguish photos from his Bullingdon Club days. "Either someone who had a wild youth can't be prime minister, so we wind up with prime ministers who never had wild youths. Or we become a culture where we simply accept people's wildness," Rourke says.

The creators of Privacy say they wish we could restore a degree of the simplicity of the pre-digital age to our complex modern existences. Rourke, who was raised a Catholic, but no longer believes, talks of "recreating a private self", and the "rebirth of mystery". Graham says he hopes we might be able to "reimagine a space where we might be private". Still, Graham thinks that a theatrical experience that entertains and informs is a beginning. "Naive as it sounds, doing a play is not a bad place to start putting it on the table," he says.

• Luke Harding's The Snowden Files: The Inside Story of the World's Most Wanted Man is published by Guardian Faber.

• This article was amended on 10 April to remove the repetition of a sentence about This House.