You are trying to fit the economic / finance model of the 20th/21st century to the 16th century and this is inappropriate at best, misleading at worst. As such, the distinction between what is a private party and what is government was different.

For example, Columbus' journeys were funded partly by the Spanish royals Isabella and Ferdinand, and partly by a private financier Luis de Santángel, who was a well known converso and finance minister to Ferdinand. But that doesn't give a picture as to how expeditions and endeavors are really funded. For his expedition, Columbus was able to extract considerable concessions based on the premise of potential success, including nobility title, military title, and financial profit-sharing. All that, even before he took sail. Why was this the case?

In the 16th century there was no fluid, reliable banking and credit system that we know today, so in many endeavors even the wealthiest royals have to rely on private parties to put up with the upfront cost, in exchange for either nobility, land grant, or some other financial obligation. This was the case for many endeavors related to exploration of the Americas.

A notable example is the Palatine - Low Countries campaign of Ambrogio Spinola in 1620s, who used his vast family's fortune to raise an army, fund its campaign northwards to the Low Countries, and had to wait years before any reward was dispensed by the Spanish crown. The strategic situation in western HRE, the Rhine, and the Low Countries, was deteriorating and the Spanish Habsburg needed to mount an offensive. But they could not do so themselves directly. They had find an agent, such as Ambrogio Spinola, to basically fund, organize, and execute the endeavor, for an uncertain promise that they would be sufficiently rewarded in the future.

This was not new. When the Duke of Alba marched 10,000 Spanish veterans from Italy to the Low Countries in the 1560s, he too had to put up with much of the cost upfront.

Nearly all enterprises of early modern europe required a private party to put up with the cost up-front. Even something as seemingly simple as sending already acquired funds to another location, requires somebody on the other side to front the money, and then be paid at a later time (or some other suitable due reward).

This practice also permeated across all levels of command and exercise. Captains are expected to pay for the needs of their men out of pocket when money wasn't forthcoming from the treasury. If they themselves get into financial trouble, then their Colonels are expected to step in. And so on and so forth.

Back to the Americas, the model followed was the the model developed during the Reconquesta, such conquistadors were given the authority to explore and conquer new lands in exchange of loyalty to the crown and fealty to uphold laws and pay taxes. Throughout the 1500s and even into the 1600s, the conquistadors were usually first to develop colonies, and the crown had to catch-up years or decades later by sending administrators and laity to impose more firm structures.

The "private sector" had a major role in discovering the Americas. Most of the conquest expeditions of the early 1500s were not funded by the Crown (in this case Castilla y Aragon) in such a significant way as Columbus had been. In fact, if you look at who pays for it, the Crown is buying into very little but instead urging close friends and patrons to invest in Columbus's endeavor. The problem with the nationstate model of conquest is that it is almost always wrong for the 16th century. By this I mean, royal funds rarely went into the exploration expeditions, but rather directed funds gathered from those newly found lands.

The issue is the contract. If we look at Spanish contracts starting in the second decade of the 16th century, we can see that the Crown rarely pays for trips to the New World. Not only that, the nature of the contract evolves over the century to give the Crown more power, but only after it is revealed how large the Americas actually are. In the Juan Ponce de Leon contract of 1512, for example, Ponce agreed to fund the trip to Bimini/Florida at his own cost. The Crown was not a part of that. What the Crown did do is offer to pay for the construction materials needed to build a fortress in those new lands, as long as Ponce directed the project. Obviously, it never got off the ground. But Florida was 'discovered' and charted officially (it had been a place known to slave raiders for a few years by then) and the Crown paid nothing for it (AGI Indiferente 418,L.3). There are other contracts, including the 1528 Narvaez disaster in Florida (of Cabeza de Vaca fame). Narvaez had to pay to ship his men from the Caribbean to the Florida coast and pay for the construction of forts and towns. As a way to pay Narvaez for his trouble, the Crown allowed him to allocate funds collected in Florida to go to the Governor, Captain-General (who could be the same person) and a warden for the fortresses. The Crown was not paying anyone, but rather allowing a remittance of cash to be kept to pay the successful enterprise workers (AGI Indiferente 415, L.1, 1526).

The most famous contract, of course, is the one that gave power to the Pizarro brothers. There are great studies on this but I would only mention that Pizarro was rather clever in agreeing to pay for passage of men and supplies to the New World (and across the isthmus) on his own coin, but was supplied by the Crown. Furthermore, Pizarro managed to get titles and periods of tax shelter that extended to himself and his brothers. This made them the most powerful political force in the New World, because they succeeded in accomplishing their contract. It also had some edits, most of which are better illustrated in Francisco Pizarro and His Brothers: the Illusion of Power in the Sixteenth Century by Rafael Varón Gabai.

These contracts and attempts at exploration, settlement, and pacification are usually done by individuals or groups of individuals. These groups are usually called companies, but do not mean business company as we might call them. However, the true private sector example is the Welser Company, based out of Ulm/Augsburg in what is now southern Germany. They were one of the most powerful trading firms in the world in the 1520s. They and their rival from the same town, the Fugger Company, regularly loaned money to the heads of state in Europe. In 1528, Charles I of Spain (Charles V of the Holy Roman Empire) gave exclusive rights to the Welsers to take Venezuela. They paid nothing for the Welsers to take control. They gave the Welsers tax shelters, allowed them to keep certain percentages of gold found, they gave any settlers the Welsers recruited tax free periods and land to manage how they saw fit. The Crown was simply allocating land that they controlled in theory (by the mercy of the Treaty of Tordesillas) to an entity that would control it in practice and eventually pay taxes and ship gold, silver, and American novelties to the Old World. By 1556, the Welser Company had abandoned their claim, but many of the Spanish settlers there remained and tried to make Venezuela something other than a Caribbean backwater. It never did reach the importance of Peru or Mexico.

So to answer your question from the Spanish side of things, 'private' individuals and groups did explore the Americas, but with contractual support by royal officials. There were some individuals who did go at it alone, but legalized their efforts after the fact, like Hernan Cortes. I would side with u/itsalrightwithme that using our own vocabulary on systems that existed 400-500 years ago becomes problematic. I have had the devil of a time trying to define the Welsers as a corporation since the contemporary German and Spanish simply call it "company" but modern definitions of a corporation fit better when explaining the quotidian experiences of the Welser business.