The term "regiment" is problematic; that would indicate a fraction of a legion ("brigade") but not a cohort (more like a battalion, about 480-600 men usually). This suggests that the term used in the original, which we'd have to see, was tagma, a Greek word meaning a detachment (tagma => "tactic", "tactical"). By the time of Titus, yes, many legions were composed of troops not levied in Italy. Those led by Vespasian, emperor during the Sack, during the civil war a dozen years earlier had been Syrians. During those years the army was made up of large constellations of legions raised in the West and those raised in Asia Minor and greater Syria. There were also large agglutinations of auxiliary troops, which in the Roman system very often kept their native organizations, arms, and fighting styles (Cretan archers, Balearic slingers, etc.). These troops usually formed up in detachments (tagmata again) sometimes called "cohorts", by inscriptional evidence (I've seen such e.g. funerary inscriptions in Britain). These units were not part of any legionary organization and were separately led and paid.
It's worth noting that in Vespasian's time, auxiliary infantry were usually paid a denarius a day, one third the legionary rate (but equal to the old legionary rate in the later republican period).
Thanks for that additional detail on Roman forces. The source material I checked out simply noted that the Italian Regiment (cohortis quae dicitur Italica) was a cohort garrisoned at Cesarea, so ten centuries of 60-100 each.