JeffB - a prize could be taken to any port, often a nearby one if it had been a 'hot action' and the prize was damaged. It would then be inspected by an 'agent for prizes' who would determine the value of the prize, both the vessel itself and any cargo and/or naval stores. The case would then be taken to the Court of Admiralty in London and a judgement made. Eventually if the prize was 'condemned' (awarded to the captors) it could be sold. Overseas there were Vice-Admiralty Courts at various naval bases. My comments above were related to the fact that home naval bases were likely to have prize agents in residence, although when the country was actually at war there would have been more agents in more ports. Bristol was a popular privateering port and a useful base in wartime.

To be able to use these mechanisms the owner of the vessel had to have been issued a 'Letter of Marque and Reprisal', authorising him to capture enemy vessels, although actions against Barbary corsairs perhaps didn't need these as they were less likely to turn up at an Admiralty Court to challenge the loss of a ship. But naval law will always apply. If not you were a pirate!

Letters of Marque and Reprisal were issued to two classes of vessel. A 'Private Man of War' was built specifically to attack enemy shipping. But any merchant vessel with a few guns for self defence could have a letter of marque in case on a normal trading voyage they came upon an enemy vessel with fewer guns. Or sometimes defending yourself against an enemy was productive, as in the case of some of the songs discussed in this thread.

Gurney - there were particular ship types used in the Mediterranean, but Western European nations used the same vessels as they would use on ocean voyages. To them a pink was a small/medium cargo vessel ship or brig rigged (3 or 2 masts) but with a narrow stern. It was this latter feature that made it a pink. They were better coastal vessels than ocean vessels because of their flat bottom and shallow draught. But there's also a different Mediterranean vessel called a pink!