It was my understanding everyone could always see them, it was just very rare to. Understanding them though was difficult.

In the opening cinematic for Gridania in 1.x you are chased by an angry Treant and are saved by a troupe of moogles playing music that soothes it. A moogle then speaks to you briefly and the Guards who found you are amazed that you can understand them, but they all can visibly see the moogles.

In ARR i thought that they just came out of hiding to assist the general populous by delivering mail because of their ability to fly as well as the linkshells failing to work. They want to help the people an it's a way to assimilate into society.

So everyone can see them but few can understand them still. But everyone knows if they give them a letter with a destination an recipient on it they will deliver it.