Given the context for my interview with Fiona Shaw, my central question – what is your favourite love poem? – doesn't seem especially tricky or prying. We meet to talk about Peace Camp, an art collaboration with director Deborah Warner and composer Mel Mercier, for which Shaw has been darting across the UK, imploring people to record their favourite love poems – and accosting well-known actors she's bumped into at airports. "Alun Armstrong! Please, will you do it?" She has recorded 570 poems in total, with voices from Cornwall, Northumberland, Wales, the Isle of Skye, and everywhere in between.

And yet Shaw is not easy to pin down. Her words keep hurtling off through exclamations, exhortations, then collapsing in laughter. She revises herself regularly, shouting into my dictaphone: "Don't write that!" She worries about anything that might come across as pretentious on the page, but in the flesh she is fast and funny – from the moment she arrives at the Guardian office in her leather jacket and dark trousers, looking like a grown-up, ultra-capable Calamity Jane. "I am excitable, as you can see," she says, "and rather big-spirited. That may be a curse as well as a blessing."

In fact, Shaw is full of another project, involving The Rime of the Ancient Mariner, Coleridge's epic poem. She is about to perform the whole text in Greece, has just finished the final run through, and is struggling, this precise second, to focus on anything else.

The seed was planted in Bel Air, Los Angeles, while Shaw was filming the TV show True Blood, in which she played the witch Marnie Stonebrook. "I used to go running every day, because otherwise you start feeling slightly desolate that you're in America, thinking, 'Why aren't I at home? What is this life and who am I?'" She would take a few verses of the poem with her as she jogged, and gradually learned the whole lot.

Her friend Phyllida Lloyd agreed to direct a performance, and suddenly this "little show, of my poem, has become one of the biggest things I've ever got involved with". Next month it goes to the amphitheatre at Epidaurus as part of the Athens and Epidaurus festival. "I spoke to a Greek journalist who was saying, 'Ah, it will be very good for Greece'. I said, 'I'm not sure it's going to solve Greece!'"

At the same time, she has been developing Peace Camp, from an idea by her long-time collaborator, Warner. Together they have tackled Electra, Hedda Gabler, Richard II, Medea, Happy Days and an adaptation of The Waste Land that was performed in London, Dublin, Paris, New York and Toronto.

Their latest project involves encampments of glowing tents – "like strange martian pods", says Shaw – set up at eight remote coastal locations around the UK: from Cliff Beach on the Isle of Lewis to Mussenden Temple in Northern Ireland, down to Cuckmere Haven in East Sussex. The tents will appear simultaneously from tomorrow until Sunday, and people can book to wander among them after dusk. A soundscape of the love poems Shaw has recorded will play over and around the tents. Shaw's voice is heard on several poems, including a duet with Edna O'Brien of WB Yeats's When You Are Old.

Will people go into the tents? "No! You park far away and walk through the fields, so you're already part of it; hopefully the tents will offset the landscape you are looking at." There was a lot of talk about which tents to use, and their connotations: "War camps, disasters, people who are sick in hospital camps – but also the way we live in little groups. And to put them on headlands means we're looking at the sea, which is our death, our life, our unconscious."

The project is part of the Cultural Olympiad, and the public have been asked to contribute their own favourite love poems, too. I ask whether Shaw included poems that have particularly affected her, but she deflects this with talk of the canon. "In the end," she says, "it's a bit like paintings, or music, or plays – there is probably a top 40 or 50 in the English-speaking world. Poems that begin to behave like boxers. They have muscle." So a recording of Shakespeare's Sigh No More has been made, of John Donne's To His Mistress Going To Bed, and Matthew Arnold's Dover Beach.

Shaw, who is 54, has long been associated with the classics. Her career started in The Rivals at the National in 1983, and she soon went to the Royal Shakespeare Company, where within six months she was talked of as the new Vanessa Redgrave. While there she played Celia in As You Like It, Kate in The Taming of the Shrew, Madame de Volanges in Les Liaisons Dangereuses.

Her stage choices have tended towards plays with poetic sweep, complicated language – and not entirely contemporary texts. She looks stricken when I mention this. "Don't say that! I think I'm part of making – I mean, I don't want to boast in this – modern theatre. I'm certainly not interested in anything old-fashioned."

But she concedes her love of difficult scripts. "I'm not at all frightened of hard words, I just get excited by them. That's not an intellectual thing – we live in such an anti-intellectual age, I've got to say that." There is more shouting into the dictaphone. "It's much more about my terror that I'm not in touch enough with feeling – and that poems put me in touch with feelings I might not otherwise get a chance, in this short life, to feel."

Shaw grew up in Cork with three brothers, an eye-surgeon father and a mother who had trained as a physicist, and had her first inkling that she might be a performer when she won a poetry recital competition, aged about 10. Her father was "very puritanical", she says, "and it wasn't good for me. He was very tough on my brothers, and very tough on me. Then we had a mother who was absolutely the opposite, a hedonistic partygoer. They were mad about each other, but they quarrelled a lot."

Her father insisted she go to university before drama school; once she'd finished studying philosophy at University College Cork, she held him to his agreement to pay for Rada. "That was very hard, but I cornered him. You have a kind of energy when you're 18, where you're so clear. And so when I got to Rada, my life's ambition had been fulfilled. I had no more after that."

Shaw has had relationships with women over the years; how did her parents react when she came out? "I don't think I ever really came out. I think they would have had no language for that." Did she find a language for it? "No. Not particularly. I mean, I had very serious boyfriends for a long time, so it's not in any way that I was brought up towards having relations with women as well as men – the word women sounds very plural, and it isn't like that in one's life. It's tiny. I've had a very modest life, I think, in every respect, and I find it hard to talk about that, for that reason. I've lived on my own for the last four or five years."

One of her longest professional relationships is with Warner: the pair first worked together 30 years ago, in an Edinburgh production of Woyzeck. Then there was a break until 1989, when Shaw was cast in Warner's production of Electra. She was 28, and her youngest brother, Peter, had died in a car crash not long before, aged 18. The production was a turning point. "For the first time – it's very late, most people have grown up much earlier than that – I think I began to see that plays and life are actually connected," she says, "and that they have a sort of healing quality … I suddenly recognised that the matter of the play, of a sister mourning her dead brother, was something I understood, and that helped me do the rest of it."

She made the films My Left Foot and Mountains of the Moon around the same time, and "had a moment where I was being bred as a movie star by America. For about six months they were really keen to get me." But she kept telling them, "I would do the film, it's just I've got to finish a play ..." She was frightened, she says, daunted: "I've probably had a lot of dips since, thinking, 'Sugar, I really missed my boat.' But it must not have been my boat. I mustn't have wanted it."

There has been screen success since, and she is occasionally mobbed in the street these days – more because of True Blood than her other recurring screen role, as Petunia Dursley in the Harry Potter films. She enjoys being recognised, "because it's made the city smaller. People smile at me a lot, and it reminds me of being brought up in a place where everybody smiled at me. It makes me less nervous. Less shy."

Beneath the noise, movement and whirlwind of work, I suspect Shaw is quiet and contained. We talk more about her current projects: the upcoming National Theatre production of Scenes from an Execution she's starring in, and The Rape of Lucretia , which she is due to direct for Glyndebourne – part of a recent, happy shift towards opera.

I return to the question of her favourite love poem. It seems impossible Shaw could have spent so long recording other people's choices without reflecting on her own. She dodges me again. "Poetry is the formal way of expressing what's really unsayable," she says, and perhaps this is the key to her rollicking deflections. If you take language as seriously as she does – and value your privacy as fiercely – perhaps few questions are more personal.

• Peace Camp will take place at eight coastal locations around the UK from tomorrow until Sunday. Details: peacecamp2012.com