Great question. Short answer: both.

There was a very powerful drive in the urban centers of the provinces to be Roman, regardless of the origins and identity of those in charge, especially in civic monumental building. The basic architectural forms had already been mostly established during the Middle Republic, in Roman enclaves in Italy, and of course in Rome itself. From those templates came the prescripted forms of temple, forum, basilica, portico, cryptoporticus, theater, amphitheater, odeum, circus, bath, etc. What I mean is that if one looks the plan of such buildings, they are mostly the same regardless of where they are, whether it be on the fringes of the Sahara, northern England, western Portugal, or Syria. This homogeneity of plan is the result of that motivation to be Roman, but we don't often know the specifics of how exactly these structures were planned or executed. Were there Roman/Italian architects designing buildings out in the provinces? Yes, in some cases certainly, and we have a small dossier of evidence of named designers, often Imperial architects, building things. But there must have also been "local" (Gallic, Numidian, Syrian, British, Iberian, Egyptian, Greek, whatever) designers who were designing structures to mimic Roman norms.

We can see evidence of this in some distinctive examples which retain something of a "local" flavor in their plan. In Britain and Gaul, for instance, there was a tradition of square-shaped sites of worship which continues to persist and results in an interesting hybrid of Celtic and Roman types. See here for lots of pictures and a good reference for this (French). These stood side by side with more traditionally Roman rectilinear temples in many places. For instance, there were fairly typical Roman temples in Britain at Colchester, Wroxeter, Lincoln, London, Bath, etc. The temple at Bath is a fascinating mixture of a very rigid "Classical" plan and very unique decoration. See here.

Despite a general tendency to "match" the plan of buildings across the empire, there is a really fascinating uniqueness in the materials and technique of construction from region to region. Every province had its own quirks in this regard. In the west, especially Iberia and Gaul, we see a distinctive style of wall construction featuring small square blocks in mortar in fairly regular coursing, called petit appareil. This is a distinctive style which appears everywhere on Roman provincial buildings in the region(s). Marble in Gaul and Spain is relatively rare until late in the Empire due to a local preference and abundance of limestone and other stones. There is also a widespread use of large ashlar block technique in Iberian construction, probably due to the influence of earlier Greek masonry techniques. Roman towns of Iberia often feature this technique on monumental buildings like theaters, city walls, and aqueducts (so see the aqueduct at Segovia compared to the Pont du Gard in Provence). Further north, in Germania and Britain, we often see building materials and techniques which are more suited for the climate and local resources, like wattle-and-daub and timber construction, especially in smaller and therefore less grand towns.

So in summary: the locals of the provinces had a vested interest in mimicking Roman forms in plan, but their distinctive identities often show in the execution and materials. For a great and exhaustive write-up, see the very excellent book by Fikret Yegül, Roman Architecture and Urbanism: From the Origins to Late Antiquity (Cambridge 2019), especially chapters 7-11.