I know this subreddit is built around detailed answers, but realistically, and with few exceptions, the answer is "they were stolen" when it comes to the golden age of piracy. Stede Bonnet might in fact be the only exception, a wealthy land owner who bought a ship specifically to go join the famous "pirate republic" after reading about their exploits.

Broadly speaking pirates started with small ships, and used those to trade up to slightly bigger and better armed ships. In fact in it's earliest days the burgeoning Nassau pirate colony would on more than one occasion take vulnerable or distracted ships with nothing more than canoes they carved from tree trunks.

As far as purpose built ships, the needs of pirates were no different than any other naval, merchant, or privateering ship in the Caribbean at the time. Pirates did run their ships and crews differently, but aside from that they weren't doing anything unique in terms of naval engagement. The Caribbean in the early 1700's was chock full of waring nations taking each others ships and raiding each others settlements.

Piracy has occurred throughout much of human history, probably dating as far back as the first sea-faring people. It happens even today, most famously off Somalia but also in places like the Strait of Malacca. It has been the cause of numerous wars (most notably with the Barbary States, but piracy happened across all of the Mediterranean for centuries before America and the colonial powers grew tired of it). It's even been effectively legalized by governments and used as a method of waging war.

So who are you asking about? Vikings? The corsairs of the Ottoman Empire, to whom most of the Barbary States belonged? The ridiculously successful Dutch privateers, such as Piet Heyn? The Caribbean freebooters, most recently the subject of some rather romanticized films? The Wokou, who plagued Chinese trade routes?

It's a huge subject. Let's drill down to a specific time and place so we can reliably answer the question.