I have the blue spot sharpnosed puffer and he puffs a few times a day. He does not get big and round like the porcupine. Instead he gets flatter and stretches up and down instead of side to side. There is a seam that runs along the bottom of the fish that stretches downward, if that makes any sense.
For reference, imagine holding an uninflated balloon and stretching it from side to side with no air in it. This is how my toby puffs. He just gets taller and deeper, but flatter from side to side. It is cool. He does it anytime a bigger fish approaches him, in a domineering way or even if a stranger walks over the the tank, especially if they have red clothing on. It doesn't seem to hurt him, but if he stays puffed for more then several seconds or gets pushed around badly while inflated, by another fish, he will fade his color and get kind of mottled for a few minutes afterwords.
Mom to the Fab 5!