From what I understand and my experience in working on this trophy, once you reach 15 Weapon Proficiency with each of the respective party members they will "unlock" the Puff-Puff scene the next time you speak with them in Accordia.
For the healslime, I am unsure how long this actually takes. I didn't actually "speak" to him in Accordia until after I had completed the game.
This is a fairly easy trophy to get, especially if you're going for platinum. Grind this either in Space-Time Labyrinth or on the "!" marked mobs. Also, using the "Fans" rank 20 weapon proficiency party skill, Decreases gold gained but increases weapon proficiency earned, will make this considerably faster. While maxing out Maya's weapon proficiency first is not required, if you're going for platinum then you will have to max everyone's weapons anyway and this party skill will make the 15-20 proficiency grind a lot faster.