Polybius explains the process pretty thoroughly actually. He was a Greek writing in the 2nd Century BC who was originally a political hostage in Rome but worked closely with the Scipiones and seems to have had an in depth knowledge of the workings of the pre-Marian army.

This is how he describes the process:

After electing the consuls, they appoint military tribunes, fourteen from those who have seen five years' service and ten from those who have seen ten. As for the rest, a cavalry soldier must serve for ten years in all and an infantry soldier for sixteen years before reaching the age of forty-six, with the exception of those whose census is under four hundred drachmae, all of whom are employed in naval service. In case of pressing danger twenty years' service is demanded from the infantry. No one is eligible for any political office before he has completed ten years' service. The consuls, when they are about to enrol soldiers, announce at a meeting of the popular assembly the day on which all Roman citizens of military age must present themselves, and this they do annually. On the appointed day, when those liable to service arrive in Rome, and assemble on the Capitol, the junior tribunes divide themselves into four groups, as the popular assembly or the consuls determine, since the main and original division of their forces is into four legions. The four tribunes first nominated are appointed to the first legion, the next three to the second, the following four to the third, and the last three to the fourth. Of senior tribunes the first two are appointed to the first legion, the next three to the second, the next two to the third, and the three last to the fourth. The division and appointment of the tribunes having thus been so made that each legion has the same number of officers, those of each legion take their seats apart, and they draw lots for the tribes, and summon them singly in the order of the lottery. From each tribe they first of all select four lads of more or less the same age and physique. When these are brought forward the officers of the first legion have first choice, those of the second choice, those of the third third, and those of the fourth last. Another batch of four is now brought forward, and this time the officers of the second legion have first choice and so on, those of the first choosing last. A third batch having been brought forward the tribunes of the third legion choose first, and those of the second last. By thus continuing to give each legion first choice in turn, each gets men of the same standard. When they have chosen the number determined on — that is when the strength of each legion is brought up to four thousand two hundred, or in times of exceptional danger to five thousand — the old system was to choose the cavalry after the four thousand two hundred infantry, but they now choose them first, the censor selecting them according to their wealth; and three hundred are assigned to each legion.> Polybius VI. 19-20

These numbers were matched by the Latin allies, with extra cavalry, and each year this process happened anew with a Roman potentially serving in a different legion each year. This applied only to citizens who had enough land or wealth; the capiti censi (literally counted by the head i.e. they owned no land) were not allowed in until the Marian reforms unless in times of extreme danger.