From what I read, it seems that Classical Greek hoplites on campaign were typically accompanied by as many attendants; Herodotos even claims the Spartans arrived at Plataea with seven helot servants each. Alexander's Macedonian army permitted a much smaller ratio of servants to heavy infantry -one to ten, and one to one for the cavalry.
Where did Roman armies during the Punic Wars and the campaigns in the Greek east fall on this spectrum? Did the Romans typically march to war with large numbers of servants in tow? To me, the way the legion was enrolled and organized doesn't seem to match the Classical Greek model -the men were selected for each legion individually, rather than the districts-based model we see in Greece. The legions had their own light troops, the Velites, who were not lumped with a heavy infantryman like the Greeks, who gave hoplites the pay of two men to cover their light armed servant as well.
Appian sheds light on your question well when he quotes Scipio Aemilianus (or Scipio the Younger, the adopted first cousin of Publius Cornelius Scipio, the son of Publius Cornelius Scipio Africanus.) addressing his men regarding camp followers, and the overall lack of discipline in the camp at around the spring of 147.
"For this reason the enemy, from the hopeless weakness in which I left him, has risen to such strength, and your labor has been made harder by your laziness. If I considered you to blame for this I should punish you now, but since I ascribe it to another, I shall overlook the past. I have come here not to rob, but to conquer, not to exact money before victory, but to overcome the enemy first,”
“Now, all of you who are not soldiers must leave the camp today, except those who have my permission to remain, and of those who go, I shall allow none to come back except such as bring food, and this must be for the army, and plain food at that. A definite time will be given to them to dispose of their goods, and I and my quaestor will superintend the sale. So much for the camp followers."
Before this, the Spanish legions that Scipio Aemilianus was given command of had a huge problem with camp followers, to the point of there being small villages around the camp, and the soldiers became overall very lazy, to the point where necessary labour just flat out was not done. While it's quite clear in nature to assume that the camp followers were not doing much to help the legion, but it seems to be quite the opposite. Instead, they were profiting from helping the legion-- which, in turn, warrants the banishing of the camp followers. Instances that the camp followers would do work for the legion could be, but not liimited to, tending to pack mules, armor/weapons maintenance, or grain storage.
At any point that the legion(s) weren't marching, their camps would have a small village of camp followers that, in Scipio's eyes, were only there for the luxury of victory without working for that given victory. However, the main reasoning that Adrian Goldsworthy gives in his book, the Punic Wars, he states that it generally made the army significantly more self-sustaining; meaning that baggage handling would be up to the soldiers, increasing morale and discipline of the troops. Thus, he expelled them and took anything that was valuable of theirs to contribute to the war effort.
Further, other examples of companions would be slaves that the Romans took following battle. However, a majority of them wouldn’t really see too much time with the legion until the Marian Reforms in 107 BC. Interestingly, merchants had gotten into the habit of following around the legions waiting for a victory or going to the legions upon hearing of their victory and buying the newly captured slaves on the spot. However, if this didn’t happen, the slaves would likely see time with the legion due to the upper command of the legion giving the slaves over to the soldiers as a part of the ‘booty.’
In later parts of the republic, after the Marian Reforms, more instances of instances more towards your question appear. The smallest unit of the legion, the Contubernium, would contain 8 contubernales and 2 auxiliary 'servants' that would assist in carrying water, taking care of the contubernium's pack mule, and often would set up the tent. Oftentimes, they would significantly aid morale by keeping the soldier's environment clean.
Sources
Appian, The Punic Wars 24
Walter Scheidel, The Roman Slave Supply.
Adrian Goldsworthy - The Punic Wars
Polybius - Histories 6