When the Romans adopted another group's deities how did it work? Were the deities officially brought in to the Roman pantheon?

You might be thinking of the evocatio, or the official calling away of a foreign deity to Rome. One of the most famous examples is after Rome’s sack of Veii.

When all human wealth had been carried away from Veii, they then began to remove the offerings to their gods and the gods themselves, but more after the manner of worshippers than of plunderers. For youths selected from the entire army, to whom the charge of conveying queen Juno to Rome was assigned, after having thoroughly washed their bodies and arrayed themselves in white garments, entered her temple with profound adoration, applying their hands at first with religious awe, because, according to the Etrurian usage, no one but a priest of a certain family had been accustomed to touch that statue. Then when some one, moved either by divine inspiration, or in youthful jocularity, said, "Juno, art thou willing to go to Rome," the rest joined in shouting that the goddess had nodded assent. To the story an addition was afterwards made, that her voice was heard, declaring that "she was willing." Certain it is, we are informed that, having been raised from her place by machines of trifling power, she was light and easily removed, like as if she [willingly] followed; and that she was conveyed safe to the Aventine, her eternal seat, whither the vows of the dictator had invited her; where the same Camillus who had vowed it, afterwards dedicated a temple to her.

Livy, V.22

We have to be careful of the specifics of this story, which is laced with supernatural happenings. Yet most scholars seem to think that that there really was a religious ceremony to bring the gods of defeated enemies to Rome. In this way, there was an official method to adopt another deity, or at least another aspect of a deity.

Yet this was not strictly regulated or controlled and the number of times an evocatio was carried out appears to be limited. The worship of other gods came and went by the will (or whim) of the people, but at times they were encouraged or banned by the emperors. For example, the emperor Elagabalus encouraged the worship of his titular god while Augustus was said to have banned foreign cults in the city of Rome.