In the earliest days of Rome, even before the city was founded there were a number of local populations with their own deities. Some of these were more evolved, for example, the Etruscans to the north or the Greeks much further to the south. More locally there were the gods of the Latin populations of central Italy such as Mars and Minerva and those which were exclusively Latin or Roman such as Juno, Janus, Minerva and Venus. In the early days of Rome Jupiter may have been one and the same with Janus.
These “original” Roman gods were a rather somber lot and so, generally speaking, more colorful and exotic foreign gods such as those coming from Greece or the Orient, made easy inroads when the Roman population came into contact with them. Different writers tell us of cities where the gods outnumbered the civilian population making it quite difficult for the individual not to make a blunder with one or the other divinity.
In fact the Romans themselves had a sort of panel of priests called the “decemviri” (ten men) whose task it was to introduce these foreign cults into Roman society.
If we wish to understand something of Roman religion it is essential to note how the Roman gods were universal whilst at the same time local to the city and its people. The universal aspect is easily understood. For example Saturn, Jupiter, Juno and Minerva all stemmed from the same Indo-European root stock and were therefore common to all peoples of the same geographical region..
As a result of this “universality” we have odd situations arising such as the capture of a Roman votive ship on its way to Delphi with a cauldron of gold. The nobles of Lipari who took the Roman ship restituted the cauldron and actually escorted it, together with the Romans, to Delphi.
The local aspect of these universal divinities was achieved through diversification of the ceremony and liturgy which might come from a common root but could be readily corrected to suit local identity, customs and culture. From here it is a relatively easy step to understand individuals sharing in public worship of the gods, as part of their Roman identity, whilst indulging in personal worship of the individual’s own gods at home.
As far as the Roman citizens were concerned the Gods weren’t confined to the heavens either but rather they could walk about the earth and be subjected to very much the same needs of other mortals (hence Jupiter’s love for disguise as a Swan I suppose).