Does Julia Gillard ever go home and spit a mouthful of toothpaste at her bathroom mirror in a gesture of disgust at her human frailty? I've watched Birgitte Nyborg, the fictional Danish statsminister in SBS's Borgen, do so, in a scene visceral enough to make the viewer flinch. And given that Gillard has also endured the odd bad day, it makes you wonder. How does Gillard reflect on her performance in the privacy of her bathroom - her own stumbles and the misjudgments? Or on the cruelty of opponents, the signs of visceral, personal hatred? How do Gillard's moments of defeat and self admonishment play out?

If you haven't seen Borgen, here's the quick rundown. Nyborg, played by Sidse Babett Knudsen, runs a minority governmen. But this is Copenhagen, not Canberra: no one shrieks hysterically about chaos and the looming end of civilisation even though deals are brokered in corridors and big reforms teeter always on the brink. Government through coalition is a thing in Denmark, like having perfect bone structure, riding your bike to work, drinking coffee at midnight and using two "Hi's" where here, one would suffice.

But while minority government might be normal, it's still crushingly hard for Birgitte. Her office door no sooner shuts than here's another person bursting in with a list of demands. The world is a challenging place, and Nyborg's Denmark is activist and internationalist. Domestically, Birgitte can barely lay one issue to rest when the next crisis presents.

Gradually, we've watched the statsminister pull her hair back: a gesture of formality and containment; a strike against chaos. And along with the power hairstyle, Nyborg has also acquired a certain quality of ruthlessness against which she has to balance her core values. Quite an assignment, that one. Her daughter, meanwhile, just wants a home-cooked meal.

Substitute a different familial obligation for the childless Gillard – sister needs a call returned, dog Reuben needs a walk, partner Tim hasn't seen me for ten days – and you see a familiar scenario playing out on Wednesday nights at 9.30pm. Two progressive lady politicians, and a whole bunch of hell.

There's much to admire in Borgen, but its exploration of female power and the responses to it set the show apart – absurd thought it is that this remains fresh for television. The subtleties are terrific. We get women's expectations of other women. (Scary). Male responses. (Variable). But best, we get Brigitte's own unvarnished attitude to her power: she wants it, even though she knows it will cost her.

This gender study is rich and revealing – but I suspect Julia Gillard would think herself lucky to be facing Birgitte's gender challenges. Copenhagen is way too cool to resist Nyborg on the basis that she's a woman. The significant resistance Nyborg faces is from her family, who would prefer she nurtured them rather than Denmark.

Gillard doesn't have that problem. Tim's on board. The extended family are delighted. If someone wants her to cook the tea we haven't yet heard from them. It's the public sphere that's proved somewhat more challenging for Australia's first female prime minister. The battle of ideas in Australia has been comprehensively clubbed by personality politics of all manifestations, with sexism one of the unfortunate undercurrents. Yep, I could imagine that Julia would prefer Copenhagen to Canberra most days.

Of course the cameras don't follow Gillard home from work, as they do Nyborg. But Borgen does make you wonder: if the prime minister could give voters an all access pass to her life, what would they see?

Would it help if we could tune in for Bruce Willis movie marathons at The Lodge, or see Gillard in her Ugg boots, or reflecting over the weeding about whether she should have knocked off Kevin Rudd in 2010? Would people like her more if they saw her having a weep in the laundry; if they could tap into the entire dimensions of the person?

Possibly. But the prime minister is one of the most private figures you'll come across in public office. Even if she could turn her life into the Truman Show, I suspect strongly she would not. Gillard is warm, and personable and charming, but she withholds. She's enigmatic. She has a quality of revealing less the longer you observe her.

This core of privacy, which I suspect fuels her famous resilience, has seen Gillard largely resist the surely inevitable suggestions that she try and "connect" with voters via the currency of the home, the personal, the family. In any case, when she has tried to play the game, the resulting "at home" ventures have seemed staged and unconvincing.

Unlike her dramatic Danish counterpart, Gillard seems to face two sets of judgments: one that would apply to any leader and another that is implicitly gender related. I wonder if people who tell pollsters they dislike the Prime Minister feel uncomfortable with a woman who feels somehow detached from them - whether the judgments she faces are harsher than those leveled at a male leader with the same temperament and instincts. (Why doesn't she emote more? Aren't women the empathisers and the binders? )

This framing of Gillard as "other" began with the infamous empty fruitbowl, and it continues as subtext to this day. Nyborg's gender goes apparently unnoticed by Danish voters and media, but Gillard's gender continues to define her political leadership, and many of the public responses to it.