The judicial part is so that the courts in the US can satisfy themselves that the terms of the treaty are being adhered to. Part of this is that US courts will also have to satisfy itself that, in the case of the request of an American citizen, that the receiving coutry is not violating rights which would benefit the accused *as if* the matter had been tried in the US. Potentially, it is at this point where doubly jeopardy - as interpreted in the US - would come into play.
There is also the matter of - did the accused actually receive a fair trial overseas - again, not by the standards of the overseas court, but by standards of right *as if* the thing had happened in the US. This is also where beyond a reasonable doubt would come into play. In the US, liberty cannot be deprived from someone without due process.
In my opinion, when a US court sees the way this has been prosecuted, with shifting theories, and with probablies being cited as "proof".... well, you get the drift.
The political part, in my mind, is where this would never lead to an actual extradition. In Canada, for instance, no extradition happens to US States with the death penalty, unless there's a guarantee that it comes off the table. Otherwise extradition does not happen.
The point being is that there's a political evaluation - is it in the best interests of the deporting-country, and does it reflect the values and political will of that country to proceed?
With the coverage this is now getting in the USA, even with a second conviction, my read is that the political stuff is in Knox's favour. For me, this is the same as saying, people actually DO know the fix was in for her and Raffaele.