This is a known bug so most likely you will have to find more than 10 nests (it took me 16 on a single save file). I'm guessing to restock weapons and manpower.
They seem to go back to their nest once there attacking power drops low. Then just follow it until it goes back to it's nest. Just sit and wait in the area and it will reappear. Note: If whilst following the pirate ship around, it disappears, don't worry. These seem to count towards this achievement. Follow this ship around and you will soon find it's nest. Just send a convoy to this area and you will probably see a pirate ship. Some times it will come up with pirate sightings - example: Last Sighting-North of Villa Hermosa. The way it popped for me was when you go into a town that you have control over and have built a town palace in, just click on it and scroll across to the tab which says 'List of Pirates'. It would then pop up on the screen and say a nest has been discovered.but this didn't count towards my 10 for some stupid reason. I use to get the info from the informer at the tavern, look for the buoy and then sit a patrol on this area and wait for a pirate patrol to enter/leave the nest. EvilKnight23 is correct in saying that the best way to do this is by following them around. I can confirm that this can be obtained with the DLC enabled. Last flag of last pirate was red, i dont know if that is relevant or not, before that I never saw a red flag I got the achievement at 5 nests discovered, 8 destroyed-13 total.
I only attacked 1 hideout when 5 where active and discovered though. Then I would wait by the hideout until a ship left then attack and destroy the hideout, then immediately search the area the ship was heading, it has a new flag, follow it to discover new hideout (make sure your convoy is strong so that the ship you are following doesn't attack you for the loot you just got from their old hideout), buy ships, (for more sailor capacity) stock sailors and weapons while passing time for pirates to gain strength, rinse and repeat. Then as I discovered them, some automatically, I fast forwarded time so they would get stronger, the best way to tell who is strongest is who has the most money reward offered. After that basically what I did was built up a decent economy with 3 convoys and established raw material buildings in 5 towns while I waited for pirate sightings.
I discovered more then 10 with patch on so i cleared system cache and declined update.