This is from an old post regarding the Passion of the Christ movie and the language of the Romans stationed in and around Judaea of the time, but it may have some relevance here. I'm not sure how well my assertion holds water - as much can change regarding archaeological research since I posted it about 7 years ago. Anyway, here was the post:
Legionaries active during the time period of the film were still mostly recruited from Italy, but it was changing rapidly. Therefore, it is very likely that most active citizen legionaries still would have spoken Latin.
Specifically in Judaea there were 3 known cohorts of Auxilia.
There were two cohorts of auxiliaries in Jerusalem and a third cohort guarded the capital Caesarea. Two cohorts of infantry and one squadron of cavalry served throughout the province.
The Cavalry cohort was Ala I Sebastenorum that consisted of Samaritans and probably spoke a local dialect and perhaps Greek. We also know of the existence of a Cohors I Sebastenorum, which was also a Samaritan unit with similar language possibilities.
Other known units that functioned in Judaea are the Cohors Prima Italica Civium Romanorum, the Cohors Secunda Italica Civium Romanorum and the Italian Cohors Prima Augusta. These are obviously Italian units and it's still likely that Latin would've been the primary language.
Of regular legionary units, Judaea at the time was considered under the command of the Legate in Syria.
The known Legions operating in the region at the time of Christ were:
III Gallica - Recruits from Cisalpine and Transalpine Gaul and likely Latin speakers.
VI Ferrata - Recruits from Cisalpine and Transalpine Gaul and likely Latin speakers.
X Fretensis - Recruits from Italy and Latin speakers.
XII Fulminata - Also orignally Gallic or Italian recruits and likely speakers of Latin.
However all Legions were supplemented at times by recruits of various regions. It is possible that any of these main contingents would've have been supplemented by local citizens in the east. So yes, there were probably some Greek speakers, but the main body of troops would've have been from a Latin origin and the tradition of language, it would seem to me, would've required new recruits to speak the most common tongue of the main body. However, in Judaea, communicating with the locals would've been easier in Greek (as a commonly known tongue among all the various parties), but there is no reason to believe that the Legions would care what was easy for the locals. An argument can be made for either side, I suppose.
Inscription evidence, letters and so forth are mainly in Latin. But only so much survives, and inscriptions and letters don't necessarily indicate what the spoken language was.