The following is modified from an earlier answer...

I know less about Columbus's voyages, but can speak to subsequent conquest attempts. There are a wide variety of experiences, but by and large the first translators were (1) indigenous and (2) enslaved by conquistadores/colonists for the specific purpose of translating during a future conquest. A common conquest method for developing translators involved luring, or outright forcing, a few inquisitive or unlucky young Native Americans onto ships during an initial, exploratory voyage. The reconnaissance voyage returned to a safe harbor, and the kidnapped Native Americans learned Spanish in preparation for their later role as translators during the entrada. As an example, Pizarro kidnapped two boys from the Peruvian coast in 1528. The young men were taken to Spain, learned Spanish, and accompanied the Conquest of Peru in 1531. They acted as translators during the famous showdown in Cajamarca in 1532 that resulted in Atahuallpa’s capture. The translators were rewarded for their service. Pizarro granted at least one of the two men, Martinillo, a share of the Cajamarca spoils. Martinillo changed his name to Don Martín Pizarro, and settled in Lima as the Interpreter General with two encomiendas to his name. Squanto/Tisquantum, of Massachusetts Thanksgiving fame, learned English after his abduction from the Massachusetts coast. He was bought, granted freedom, and spent years attempting to return to Massachusetts, arriving home right before the Mayflower made landfall.

Shipwrecked or captive Spaniards were valued in Native American communities as translators and their ability to provide insight into European aims. As an example, in Florida the de Soto entrada encountered Juan Ortiz, a Spaniard captured while searching for the lost Narváez expedition. Ortiz learned the Timucua language during his years in Florida and served as a translator while de Soto rampaged through the region. De Soto said of Ortiz, “This interpreter puts new life into us, for without him I know not what would become of us.” Kidnapped Native American translators could also escape their captives, and return to their homeland armed with new languages and valuable insight into European objectives. Don Luis de Velasco, a Native American abducted from the Virginia tidewater region in 1561 returned in 1571 with Franciscans establishing a mission near the James River. Don Luis escaped, and returned with an armed party that killed the Franciscan fathers.

Even when clunky, Native Americans and Spaniards used any means possible to communicate. The Conquest of Mexico famously required two translators. A shipwrecked Spaniard who lived for eight years among the Maya, Gerónimo de Aguilar, translated Spanish to Maya, and Dona Marina/La Malinche, a Nahua noblewoman from the frontier of Nahuatl-speaking central Mexico, then translated Maya to Nahuatl. The clunky Nahuatl to Maya to Spanish translation was abandoned once Dona Marina/La Malinche learned Spanish and could directly translate from Nahuatl to Spanish. Her constant presence near Cortés eventually led the Nahaus to dub Cortés Malinche, “as though captain and interpreter were one”. As a final aside, I’ll include a brief mention of the many Catholic missionaries who worked with Native American linguists to learn indigenous languages and translate religious documents. Works like the Castilian-Timucuan catechism and confessional published in Mexico City in 1612 constitute the earliest surviving text of a North American Indian language.