There are two kinds of opposing pirates in the game: generic ones and "named" ones.
Generic pirates are a nationless type of ship that can provide even law-abiding players (those who have decided not to make any of the four nations angry) some revenue. Pirate Raiders seem to be most common, who are generally out on missions to attack cities, but there are also pirates that just prey on shipping, and also priveteers who work for individual nations. I think I remember seeing Pirate Smugglers as well. Attacking them can pay off, but probably not very well.
Named pirates are a different breed entirely. They are the nine other people on the 10 Most Wanted list in the subscreens. They all seem to follow their own piratical itenaries, and as the game continues their standings on the list increases, although I don't know if that's a result of their AI program's success or if it's faked.
Unlike you the player, who despite the danger of his profession has no way of being permanently evicted from the game other than retirement from old age, if you defeat a named pirate, he vanishes, although his treasure remains if it's not been found yet. (I've yet to find a named pirate's treasure, so more info on that later. The manual says if you find a pirate's treasure, he'll come looking for you.)
Named Pirates can be difficult to find, it seems: played 15 game years without even encountering one. Their bases seem to be selected from the randomly-placed Pirate Haven settlements at the start of game, and both named pirates I've taken out so far have been nearby. Hints as to which pirate is in which settlement are found in taverns, but only those near that pirate's base.
Named Pirates are worth large rewards if defeated: the one I just took out was worth almost 20k in gold. It must really sting to accidently sink one.... Named Pirates are always significant opponents in swordfights; the one I just beat was the first opponent in the game that I actually *had* to use a Taunt on, and I've beaten Montalban twice so far. A FAQ on the game claims that it's possible to win a pirate battle through demoralizing his crew, but that it's extremely hard to do so.