The first professional soldiers in Italy not tied to the traditional Urban Council-Vassals of the Urban Councillors arrangement had actually appeared in the thirteenth century after the Crusades, when Germano-Frankish soldiers returning from the Holy Land were left stranded in Italy. Often little more than roving bandits, they were often rounded up and put to good use during dynastic infighting, and eventually lots of them put themselves in positions of serious power (see, "The Other Norman Conquest" by J.J. Norwich).

Subsequent mercenary companies were more often than not veterans of fighting in France and/or Germany, where dynastic fighting had blurred the lines who precisely was fighting against who, and individual vassals and the forces they commanded opportunistically backed different sides based on plunder, risk/reward, and who seemed to be winning. The first "famous" mercenary companies came from places peripheral to the larger conflicts of the time composed of men far from home with little to loose and everything to gain: Catalans, Basques, and Bretons followed by English, Welsh, and Burgundians. Most early companies were mounted men-at-arms; men who had been born into landowning families, but perhaps stood to inherit little more than a horse and a name. These men were quick to attach themselves to powerful landed lords, offering their service in battle in exchange for lodgings and a place in the court. Following their lords to war, many stayed behind to plunder and offer (or extort!) their services to whomever could offer them the most.

Many of these companies had slowly trickled into Italy in the 14th century, being employed first by the Marquis of Monferrat in northwestern Italy (unsurprisingly near France). Although the Visconti of Milan were also early adopters, as after seizing power in Lombardy they rapidly expanded their holdings.

The later appearance of Italian mercenaries can be attributed to a sort of snowball effect. At the late date of 1377, Alberic of Barbiano established the first Italian-only mercenary company. Alberic himself was a son of the Count of Cuneo, a vassal of the Marquis of Monferrat, and was most probably present when his father was called to arms in the conflict between Monferrat and Savoy. Instead of returning home at war's end, Alberic, then sixteen, joined John Hackwood's White Company, a company of English veterans of the hundred years war who had been employed by the Marquis.

The White Company was employed first by the Republic of Siena, and later the Papacy. When the Pope, in the pocket of the King of France, was convinced to move to Avignon, Hackwood found himself putting down revolts as cities under Papal control tried to assert their independence. Florence, Milan, and Padua also raised armies and attempted to redraw their borders.

After witnessing Hackwood's forces raze the rebellious city of Cesena to the ground, Alberic left the company. Disgusted by the unchecked antics of foreign mercenaries, as well as with the Papacy which at this point was already a pawn of foreign powers, Alberic pledged his services to Barbanò Visconti of Milan, rounding up the Italian mercenaries who would follow him. Alberic's Company of St. George, as they came to be called, would serve as an incubator for many mercenary captains of the next century.

Legend has it that during these wars in central Italy, a young man called Giacomo Attendolo was ploughing his father's field when a group of knights in the service of Alberic of Barbiano rode past. Dismissing the absurdity of a landowner's son ploughing a field (although Italy was recovering from the plague. Shortages of manpower makes it more plausible than you'd think), the legend goes that young Giacomo, bored with country life, decided to throw his plough into a tree. If it fell back to the ground, he would get back to work. If it stayed stuck in the branches, he's join the company. The plough stuck. Giacomo Attendolo rode after the knights with his horse and his arms.

The story might be questionable, but Giacomo Attendolo did become one of the most celebrated mercenary captains of the early 15th century, first under Alberic of Brabiano and later with his own company, first in the employ of Perugia, and later becoming Cheif Constable of the King of Naples. In his career, he would earn the nickname Sforza, or "Strength", and his son, Francesco Sforza, would not only become the greatest general employed by Milan, but would also marry the Duke's daughter and only child, inheriting the throne of his father's first employer.

So the TL;DR is that the circumstances by which a company would be founded were varied, however it was definitely not a regulated process.