Using the Romans as an example (largely because I did a ton of work in NT/Early Christianity for a few years), it varied. Certain areas of the empire were under less control than the others. Take, for example, the province of Judea, where it's often said that Roman careers came here to die. The Roman prefect/procurator (it changed in the mid-60s, I think, see Sherwin-White's Roman Society and Roman Law in the NT) was nominally the head of the colony. He made the decisions. In certain cases, he had the authority to have people arrested and sent to Rome, c.f. Colin Hemer's The Book of Acts in the Setting of Hellenistic History.

However, the governor did not rule alone in most cases. In the case of Judea, he had a series of client kings (e.g. Herod the Great, Herod Antipas, Archelaus, etc.) who ruled their own lands for administrative ease and (often) in an attempt to be sensitive to the needs of the subjugated people. These rulers would often oversee some of the more tedious, day to day aspects of administration, leaving the governor to deal with much more significant issues.

I can't commentate as to border control, but as to your remote village question, yes, absolutely. The most notable example I can think of is the Qumran community, who, while nominally under Roman rule, seem to have more or less done its own thing for the better part of the 100+ years until the Romans destroyed the site (c.f. Magness' The Archeology of the Dead Sea Scrolls or De Vaux's Archeology and the Dead Sea Scrolls.