The post-Calamity post moogle movement began at the behest of Kan-E-Senna and the Seedseers. Prior to the Calamity, moogles were exclusively, albeit incredibly rarely, seen in the Twelveswood. Even many Gridanians had gone their whole lives without ever seeing a moogle. They're very reclusive creatures.
After the Calamity damaged all linkpearls and left the city-states without a fast means of communication, the Elder Seedseer asked the moogles to come from hiding and work as couriers between the city-states. Now, according to a later mail-carrier quest, these moogles still deliver their mail without being seen. They turn invisible and drop it off where the recipient will find it. The senior mail carriers who hang out in the city-states, however, remain visible near guarded areas of the city-state to receive letters.
Why are moogles so reclusive? Well, one reason is that pre-Calamity, Ul'dahn traders would try to steal stray or injured moogles and take them back to Ul'dah to sell their rare pelts and poms.
But to answer your original question, moogles can choose to be invisible at will. If one reveals itself to you, it either trusts you or it was a clutz and forgot to hide itself, which happens in the moogle quests. They're not exactly the most graceful creatures...
After the Calamity damaged all linkpearls and left the city-states without a fast means of communication, the Elder Seedseer asked the moogles to come from hiding and work as couriers between the city-states. Now, according to a later mail-carrier quest, these moogles still deliver their mail without being seen. They turn invisible and drop it off where the recipient will find it. The senior mail carriers who hang out in the city-states, however, remain visible near guarded areas of the city-state to receive letters.
Why are moogles so reclusive? Well, one reason is that pre-Calamity, Ul'dahn traders would try to steal stray or injured moogles and take them back to Ul'dah to sell their rare pelts and poms.
But to answer your original question, moogles can choose to be invisible at will. If one reveals itself to you, it either trusts you or it was a clutz and forgot to hide itself, which happens in the moogle quests. They're not exactly the most graceful creatures...