These two papers
Ross Street, Two-dimensional sheaf theory J. Pure Appl. Algebra 23 (1982), no. 3, 251-270, MR83d:18014, doi
Ross Street, Characterization of Bicategories of Stacks, MR0682967 (84d:18006), p. 282-291 in: Category theory (Gummersbach 1981) Springer LNM 962, 1982
basically supply (via observations by Mike Shulman, see StreetCBS) definitions on the $n$Lab of regular 2-category, coherent 2-category, exact 2-category, 2-pretopos, 2-site, 2-sheaf, and Grothendieck 2-topos, as well as the proof of the 2-Giraud theorem. The definitions we are interested in are the bicategorical ones, which appear in the second paper, but the only proofs are in the first paper which does a strict version for strict 2-categories. Note in particular that where he uses the prefix “bi‑” we use either “2‑” or no prefix; thus our 2-sites are what he calles bisites, etc. There are also a couple of mistakes in the definitions and proofs which we have tried to correct in the relevant $n$Lab entries.