Alright, here is my theory (spoilers are coming, so mind your step!):
Osgood is... HUMAN!
From what we have seen of this species, Zygons require to keep originals alive to be able to shift into their shapes. Not only that, they must keep them alive at least long enough after cloning them so as to learn to immitate habits, interrogate them and access their memories.
If Zygella – or Bonnie, hahah! – was able to take the form of Osgood, that means there IS an original out there, alive, to make a copy from. HUMAN, ORIGINAL and ALIVE :D!
BUT... this theory only works if the afore mentioned MO is the ONLY way for Zygons to make copies.
The counter-theory includes an alternative where Zygons ARE able to copy another Zygon already in the shape of someone (or something) else, not only being able to preserve body prints after the originals are gone (or killed) but also able to reproduce such body prints (making several copies without the originals).
As for why the Doctor finds the question important, I agree with Kerr Avon:
"[...]the subtleties of the reaction to the question and the response itself may reveal more to the Doctor about the character of Osgood which ultimately leads to the question/offer that only the special and extraordinary hear."
The Doctor may have been testing her, somehow? So to speak.
All in all, and for as much as I do crave to know... the truth is we SHOULD NOT CARE. It's been said before and I agree. "The Osgoods" are the whole point. There being two of them, living like hybrids (sort of) and not letting anyone know which one belongs to which species, makes them THE ULTIMATE peace treaty. The ultimate BREATHING peace treaty. In a box. Of flesh. A twin box... I have to stop.
PS1. I'd like to take a moment to acknowledge the love these two sisters developed for one another –
and the hurt of the surviving Osgood resulting from the other's death.
= = = = =
To tip the scales a tad further, there is also the fact that the Zygon cloning process involves body prints taken TELEPATHICALLY. Which leaves the doors open for the possibility of Zygons being perfectly capable of sharing information and copying body prints off another Zygon.
Additionally I have read a third alternative theory on Facebook,
of there being THREE Zygons from the beginning, keeping an original human Osgood in a pod with two Zygons roaming the streets in Osgood form and all three telepathically linked, sharing information regardless of location... clever!
I have to admit it IS far fetched but most interesting!
PS2. In any case, for as much as I do want her to be the human, original Osgood – because we loved her and there is loss in her death even if there are copies of her left to spare, plus poor Kate... ah the feels T_T – as others have already stated before, it makes for a stronger, way more powerful statement for her to be the Zygon Osgood. So devoted, committed to peace, and so hurt by her human sister's death. I've used the terms before: THE ULTIMATE BREATHING and walking peace treaty.