This was done all over the Empire - which is why I mentioned Egypt. Of course Egypt was ruled by the Ptolemies, who WERE Greek, and the Romans kept the Ptolemaic coinage system that was already in place. But you'll see Egyptian gods on Egyptian coins, British gods (Sulis/Minerva) on British coins, etc. But the official administrative records would still be in Latin.
If you care to puzzle through this very weighty review of a German work in the Bryn Mawr Classical Review (a collection of essays on colonial administrative practices in the Empire), it appears that the local proclamations, inscriptions, etc. were bilingual and that local governors did operate with reference to existing local law. But the structure of the Empire remained Roman and Latin, and internal records (of which few survive) appear to have remained so as well. I imagine that Greek in the east became more and more the language of empire as East and West split, but not as early as 33 A.D.
Greek was spoken in the east for hundreds of years - Latin would not have been the language of any local there in 33.A.D. earlier or later.