The Navy have a 10 piece puzzle to collect called Stewart`s Puzzle, while the Pirates have a 10 piece puzzle called Joe`s Puzzle (different to the Treasure Seeking 50piece one they also have).
The first two pieces of this puzzle can be done by sinking treasure ships for your faction leaders, you should complete the sink bouy quests before you will be given these quests.
The last eight pieces of the puzzle can be gained from capturing the various islands on the map, the sequence goes in order as described in the island capturing topic, when you are standing inside the flag stand room you should see a chest. … or in some cases two chests.
The blue colored chest is for the Navy while the silver and green chest is for the Pirate League, neither chest should open for someone from another faction.
Use the puzzle piece from your inventory to add it to your puzzle collection, once you get all 10 pieces you will automatically be given your reward.