During the reign of the Catholic Kings and Fernando's hold over Spanish power until Charles I was of age, the placement of notaries or escribanos was based on education and merit, for the most part. As the Catholic Kings (Isabel especially) wanted an administration and royal bureaucracy that could work efficiently without significant oversight from the Crown, having educated, intelligent, hard-working officials was key to their success and consolidation of power in Europe and elsewhere. The post of notary was therefore given to promising young lawyers, legal scholars, or those hoping to follow their family into the world of royal administration.

Charles I kept this system in place as much as he could. In the Americas, however, the major issue was that for political and administrative posts, one had to open the qualifications up a bit. If one was a learned individual, they probably stayed in Europe during the first half of his reign. Those feisty individuals who left Europe with education and personal drive, usually did so because their low birth or connections could not guarantee them a successful career opportunity. Hernán Cortés, for example, had a notary in his family and did receive some education in the law and military matters. Upon reaching Española, he earned the position of notary for a cabildo west of Santo Domingo. After working to improve his connections, he joined an expedition to pacify Cuba and he was made a notary of the new colonial capital of that island. He earned his way to better positions just as Fernando and Isabel had always intended.

This system was not to remain in place for long. By the end of Charles I's reign, it was common for some posts to be handed out without regard for education or experience. In the borderlands areas, just the ability to be Spanish, male, and alive at the same time could give you the position. Once Philip came to power, the situation was made more efficient, but produced less qualified candidates as well. Small offices, such as notaries could be sold to those with ready money. This usually meant individuals of social standing and political power. Often, the elite descendants of conquistadors or middling elites from Spain. Philip was in debt and having elites from Spain pay to work for him meant that he not only got to access to easy capital, but he also consolidated Spanish power in the colonies over the conquistador families who had created their own social class for themselves. They had political careers to cultivate and becoming a notary was a foot in the door of imperial and colonial administration. The cabildo notaries were the first to be sold off, followed by those attached to courts and governmental bodies. Other offices soon followed suit, restricting the eligible population from those educated and intelligent enough to handle administrative work to those able to pay and later to those able to pay and having been born in Spain. (There has not been enough done on the sale of offices as investments. It's a big project, I would imagine, but I would be interested to see how it was handled.) From what Haring provides, the price for escribano positions ranged from 50,000 to 60,000 pesos in the 17th century.

So, to answer your question: Governors and their secretaries could appoint nearly anyone to serve as a notary, although it served the best interests of the early colonial world if that person was able to do the job. Only late in Charles I's reign was ability seen as less important than status and political capital. By Philip II's reign and to the end of the Hapsburg period, money spoke volumes over ability.

Haring The Spanish Empire in America 150-155;270-271 (It's an older source, but one of the few that scholars should reexamine.)

I did a quick JSTOR search and saw that the first significant article is J. A. Robertson's "The Office of Escribano in Sixteenth-Century Havana" in the Hispanic American Historical Review, Vol. 49 (1969). I think it will be of much greater use to you.