I know that many player required solo servers or PVE server, because other players had sunk there ship and stolen their goods. It is happend to us aswell, and it can be so fustrating after 6 hour of athena run....
Anyway I know that as well that turning of the PVP is impossible, and the private servers are not a good solution (too expensive, and the world events like the kraken would break the experience). My solution is different. I would suggest to implement an infamy/honor system. Every pirat would have statistic that shows how many "crime" did they commit against other players. Whit the current log system in the game I guess it would be possible to track such event as stepping on other's ships, shoot with cannon to other player's ship, etc. These events would make a pirate more and more infamous. While events such as just passing by player ship, without trouble, or giving treasuer to others, sharing a drink would lower the infamous repution.
And with this system, you should be able to seperate the players between the servers. So the really hardcore pirets who like to steal from eachother, and fight with eachother could be together, and those are more peacefull, and try to avoid the other ships, could be togehet on a different server session. So every player could enjoy their onw playstyle as they are amoung similar players.
(Sorry for any mistake in my english)