Proof of Giroux's correspondence

As far as I know, there is no publicly available written proof of uniqueness. Goodman's thesis pointed out by Chris proves neither uniqueness nor existence. What he did was to provide some of the first steps towards understanding the link between open books and tightness. Before that, he does sketch a proof of the open book theorem but, if I remember correctly, this sketch contains less information than what Giroux wrote in the ICM proceedings. In particular it entirely fails to cite Siebenmann's paper that Giroux cites twice in his uniqueness sketch and is the crucial starting point. This paper has been very hard to find for 30 years, but eventually got published as
Les bissections expliquent le théorème de Reidemeister-Singer: Un retour aux sources
Annales de la Faculté des sciences de Toulouse: Mathématiques, Série 6: Volume 24 (2015) no. 5

I'm almost certainly the mysterious person that Anubhav Mukherjee mentions in his comment, but writing a proof of this theorem is way beyond the scope of a mathoverflow answer, I'm sorry. I could probably answer more specific questions though.

This might suffice for you, it is not published and only slightly longer than Etnyre's sketch, but without exercises. This has been shown in the PhD thesis of Noah Daniel Goodman (a student of Etnyre), specifically Theorem 3.4.4:

Contact Structures and Open Books