No, Roman rule in its empire did not just benefit people living in Rome or in Italy. Instead, it was rather beneficial to most of the people in the areas that were conquered. This is one reason that the Roman Empire was attractive to many people and why people like Germanic tribes often wanted to move into areas controlled by Rome.
One of the famous periods in Roman history is the Pax Romana. During this time, Rome was so dominant that conditions inside the empire were very good. There was little social upheaval. Travel within the empire was generally safe (and generally easy as the Romans built really good roads for military reasons). This was good for everyone. People did not have to worry about having their lives disrupted (or even ended) by wars or rebellions. They did not have to worry as much about being robbed. They were able to benefit from trade because they could get things that were not available locally. They also benefitted from things like the water supplies brought by Roman aqueducts.
In many ways, the people who were conquered by Rome often benefitted in many ways from being subjects of Rome.