At the beginning of Privacy, James Graham’s 12th play and a transfer to New York’s Public Theater from the Donmar Warehouse in London, a disembodied voice informs theatregoers that, contrary to protocol, they are to switch their phones on and connect to the Public Theater Wi-Fi, the password for which is beamed on to a screen at the back of the stage. This, says the voice, is because there will be an element of digital “audience participation”, news of which provokes much chuckling but which I know for a fact makes every British person want instantly to run and hide in the toilet.

It is the stirring of this anxiety – of being watched, revealed, potentially exposed to ridicule – that is the brilliance of Graham’s play, a piece of docu-theatre that combines actual interviews with straightforward drama and a hugely innovative set of interactions with the audience.

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It is also something of a love story. The protagonist, identified merely as The Writer and played by Daniel Radcliffe, is a proxy for the playwright who, in the wake of a painful break-up, is trying first to use the internet, then to wean himself off it, in order to make meaningful connections with others. Through his personal journey, Graham gets at the play’s real purpose: to animate the post-Snowden world in such a way as to make us, the audience, feel that not only are our intellectual but our emotional liberties at risk.

At the centre of the play is an understanding most of us grasp at but have yet fully to articulate: that the more we share with strangers online, the less available our interior lives become not just to our intimate relations and friends, but to ourselves. The play opens in a psychotherapist’s office in north London, where The Writer is meeting his therapist for the first time to discuss his state of emotional paralysis. What is the introvert to do in an age in which the sharing economy – emotional, moral, financial, structural – is the only currency available?

Facebook Twitter Pinterest Privacy features audience participation via internet connection. Photograph: Joan Marcus

Radcliffe is such an affable actor that you ride happily with him through the archer moments of this set-up and on to New York, where he travels to find his ex and interview a lot of people with theories or experience of internet culture, among them the academics Sherry Turkle and Jill Lepore, the Guardian’s Ewen MacAskill, who broke the Snowden story, and titans like Eric Schmidt and James Comey, the director of the FBI, with whom The Writer imagines meetings. This roster of characters is played on rotation by Rachel Dratch, Michael Countryman and De’Adre Aziza, excellent all, and sets up the question: who is at fault here? The Writer, for shutting himself in his room and refusing to “participate”, or the culture around him by which he’s judged wanting?

There is a sense at times that, in order to answer this question and shore up its thesis about “pathological” intrusion, the play’s net is thrown too wide, roping in Amazon drones, Uber Pool, spam about penis enlargements and a bank’s automated messages, as well as eavesdropping technology and NSA law-breaking.

Privacy debuted at the Donmar two years ago and despite some fresh (and good) gags about Brexit, it also occasionally looks its age. No one in 2016 can get away with a critique of the internet that mentions the banality of people uploading pictures of their lunch, and the hoary old story of Target knowing when a woman is pregnant before she does is, at this point, widely treated with skepticism, but appears here uncritically. At its worst, the play can feel like a primer for an 80-year-old starting to get his head around this thing called the internet.

What saves it is the second half, when these disparate elements of online culture are pulled together by Graham and presented as a universal threat to our very notions of personality. In what is, for the most part, a funny, smart, analytical play, the end is profoundly moving and even shocking. If the stuff that makes us us is being scattered to the four winds or handed over wholesale to Apple, what’s left?