Like in the Greek sphere, the Roman chariot teams were originally owned by private individuals, necessarily elite men with a lot of resources. But by the time of the Second Punic War (see Kyle for this dating; no real evidence until the 1st century BCE), racing had become big business, and a more comprehensive and efficient arrangement was required. From this point the teams were owned by business entities called "factions" (factiones). They are similar to modern corporations, with multiple shareholders or controlling interests (uncreatively called domini). The evidence is not great for the Republic, but it seems to have been, like most business, primarily the domain of the equestrian class rather than the senatorial class. (note to beginners: "equestrian" here has nothing to do with race horses; it is just the term used for Rome's business class).

The factiones controlled virtually every aspect of the race, including the animals, the equipment, and the jockeys, and also owned their own stables (stabula factionum) and headquarters a short distance away from the Circus. By the Imperial period, the factions were vast syndicates with agents and scouts in every corner of the empire, racing multiple teams at Rome, Aquileia, Mediolanum, Carthage, Sousse, El Djem, Thugga, Utica, Alexandria, Antioch, Caesarea (Isreal), Aphrodisias (Anatolia), Corduba, Tarragona, Arles, Lyon, Vienne, Trier, Camulodunum (Colchester), and other locations I am forgetting. They sourced horses from Spain, Numidia, Thessaly, Anatolia, and Persia, typically from specialty ranches, some of which were Imperial estates. The factions were private armies, essentially, with hundreds of animals and thousands of support staff: jockeys, of course, but also doctors, handlers, coaches (hortatores), trainers, groomers, wheel-wrights, water-sprinklers (sparsores), enforcers, accountants, slaves, and every other possible chariot-related job you can imagine. There were internal hierarchies for all of these, including the jockeys. Beginning in the Imperial period, we start to see front men for the factions, almost always famous jockeys. These were called domini et agitatores and the faction leveraged their fame as figureheads. They were nominally the face and dominus of the faction, but of course the money-men behind the scenes still ran the show. Though in the Imperial period the Emperors frequently meddled with the races and their workings, the factions remained "independent" entities until the third century CE or later.

Cameron is still the most comprehensive study on this topic: Circus Factions: Blues and Greens at Rome and Byzantium (Oxford, 1976).

See also Kyle, Sport and Spectacle in the Ancient World (Wiley-Blackwell 2015); and Alison Futrell, The Roman Games: a Sourcebook (Blackwell 2006).