There were black market merchants, smugglers, counterfeiters, and launderers of goods that they could sell it to. Smuggling probably being the largest one. Not every port was secured or had a harbormaster who cared (bribed) enough against pirates sailing in to sell their goods. If it was, pirates could offload their goods ship to ship or at a beach site, to a more legitimate seeming ship or a smuggler. For goods that did not have any system in place to verify their source, they could simply be sold to any seedy merchant. For goods and ports that were controlled, a more sophisticated system would be needed. The process being sort of similar to modern day money laundering or cigarette smuggling. Pirates takes possession of the goods, then they must get it in condition it can enter the legal market or find a black market buyer. This was usually done by a 3rd party counterfeiter, forger, or merchant. The goal being either to a final consumer who was willing to accept it as black market, or to make it appear legitimate for a consumer who only wants a legitimate good. Piracy was only a small part of the capture/transportation of illegal goods. Many illegal goods began legitimate or were stolen, and became illegal simply because they were smuggled to avoid taxes and duties (or in some cases, only a select few had the rights/privileges of trade).

The respective nations were not incompetent and enacted a plethora of measures to combat both piracy and the trade of illicit/pirated goods. Even in days before electronic monitoring, UPC codes, etc, a lot of goods required stamps, marks, and certification from both goods and merchant. Some measures were somewhat counterproductive, like Spain's mercantilism policies which meant the colonies could only trade with Madrid. This created a black market demand for other European goods.

For some periods there were some less than lawful ports like Nassau and Tortuga which served as pirate havens and markets for illegal goods. More ports fell into a more grey area, where most was legitimate, but some illegal goods were traded. Many more towns and ports were known for smuggling than outright piracy. Cawsand, England was one of them. As time went on and the bureaucracy expanded and the nations and companies had more resources and funds at their disposal they were able to chase pirates out through force and inventory controls.

The actual process of smuggling, mercantilism, and the combating illegal goods would be a large response to questions on their own.

Sources:

Transportation Act 1717

Piracy Act 1698

The History of Pirates Dr. Angus Konstam

Under the Black Flag: The Romance and the Reality of Life among the Pirates David Cordingly

Smuggling: Contraband and Corruption in World History Alan Carras

http://www.oxfordbibliographies.com/view/document/obo-9780199730414/obo-9780199730414-0077.xml

http://americanhistory.oxfordre.com/view/10.1093/acrefore/9780199329175.001.0001/acrefore-9780199329175-e-263

http://www.oxfordbibliographies.com/view/document/obo-9780199730414/obo-9780199730414-0197.xml

Pirates were pretty well known for being hot-blooded (mostly) young men who hit the streets of Port Royal with their mates and spent a whole lot more than a few shillings "giving themselves up to all sorts of debauchery with strumpets and wine." So it might be kind of a surprise that pirates' views of both what constituted loot and what to do with that loot were quite practical.

First, there was always the possibility that the captured ship itself was a prize. When John Duglas captured the nine-man ship Blue Dove in 1664, it wasn't nearly good or big enough for his own crew, and the pirates simply locked the other sailors belowdecks and ported away the cargo. But saltwater was hard on the hulls of even good ships, and trading up in terms of size, weaponry, and condition was always tempting. Humphrey Thurston abandoned a ship still worth 300 pounds as an unsalvageable wreck (in the Caribbean, where vessels sold for much lower rates than in England) when he captured a Spanish man-of-war in 1670.

Although spending by privateers doubled the amount of alcohol sold in Jamaica in the 1660s, pirates were keen to cut out the middleman and use captured cargo as their source of food and liquor. The governor of Jamaica in the late 1660s even complained that privateers never bought actual food from his merchants and farmers! Modyford was grumpy about this, but land-based owners of semi- or not-actually-at-all legit privateer fleets were rather happier that they only had to supply breadstuffs--if that! Pirates intending on a long period at sea would butcher live animal cargo and salt the meat to preserve it for months of use. Sadly, they also killed, salted, and ate the dogs who kept sailors company and helped with pest control on ships.

But the most common fate for captured loot by far, of course, was basic economics: you have product, you sell it. In the heydeys of Caribbean piracy, the sale of booty in pirate-heavy towns was one of the most important factors shaping local day-to-day economy (not just sweeping sugar trade-driven trends). A single, lucrative haul of log-wood in 1667 crashed the price of the commodity in Port Royal for four months. More typical were steady low prices, locally, of things like cocoa and indigo that were sold dog cheap (or rather, "dogge cheape," because early modern English loves you and wants you to be happy).

This was a boon for local traders, of course, who went right on to sell those commodities elsewhere for a significant markup and profit. I have to mention that this was unfortunately true especially for captured slaves. For some reason, local plantation owners had no extra troubles when they treated captured pirates who had angered them as slave labor, but slaves seized off ships were called mischief-makers and sold off-island. I've seen it suggested that pirate crews may also have retained specialized slaves for their own use and profit--pearl diving was a specific and valuable skill, and slave-divers were sometimes seized off attacked ships.

Pirates dumped their merchandise-loot into local economies, driving the development of major towns (and also smaller ones, to evade local governments' attempts to stick a hand in their profits). The cash flow back out, even with the dirt cheap prices, should have been enormous. But pirates were the live fast die young sort (a study of 12 privateer captains identified in sources from 1662-63 and again in 1670-73 found that after just a decade, 7 were dead, 1 was in jail, and 4 were petty laborers). And so all that money they received from selling their loot went right back into the pockets of the locals who had exchanged it in the first place.

...Only to, you know, complain bitterly to anyone who would listen how frequently they were the target of raids from pirates whose way of life, after all, was stealing.

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The answer, basically, is that they resold it. The more complicated answer is - it depends on when they were operating.

The Golden Age of Piracy (the periodization most historians use for the Jack Sparrow-y, largely Caribbean explosion in piracy) was split into two phases by events in Europe. By the mid 17th Century, France, Spain, and Great Britain were all expanding colonially and competing for space in the Caribbean. In these countries were often embroiled in costly wars against each other in Europe which largely monopolized their navies. The result was that the profitable shipping coming out of new colonies in the Americas was relatively unguarded. All three countries, but especially the British, took advantage of this by hiring privateers. These were essentially pirates hired out by one of these governments to attack the shipping of their geopolitical enemy. They would be paid a commission for ships captured and were also allowed to fence seized goods in the colonial holdings of their patron power.

Now in the 1720s, this series of European wars finally ended, and for the first time in a while, there was peace. Peace allowed the transatlantic trade to really boom, but it also eliminated the need for the colonial powers to hire out thugs to attack each other. By this time thousands of privateers were operating in the Caribbean. The pickings were just getting really rich, they weren't all going to disappear just because some wig wearing colonial governor told them to. Some pirates did try to go straight, but they often found that life as an honest sailor could be demanding. The mortality rate for merchant seamen on the now flourishing triangular transatlantic trade route was often higher than the slaves they would sometimes be transporting. The result was that even though piracy wasn't as profitable as it used to be, it still remained immensely popular. Many of these pirates set up their own "pirate ports" or "pirate republics" (like Tortuga in the Pirates of the Caribbean). The largest of these was at Nassau. There pirates could re-provision, re-arm, sell their goods illegally, and rest between plundering.

Sources: Peter Earle, The Pirate Wars. Gabriel Kuhn, Women Pirates and the Politics of the Jolly Roger.