In Tibetan Buddhism there is a sect of monks who specialize in "Telepathy". Their job essentially is to protect the Tibetan people from natural disasters. Does anyone know the process on how these Tibetan monks are chosen? And why are not all worthy Tibetan monks allowed in this position?
Not all worthy (enlightened) monks have psychic powers. In fact, Satan or Mara also has psychic powers. The Buddha's enemy Devadatta, who tried to murder Buddha, also had psychic powers. Psychic powers, such as telepathy, can only be developed by individuals that have a latent disposition for them. The Buddha had many fully enlightened disciples but they all did not have psychic powers. The most lucid (Sariputta) did not have psychic powers. The disciple famed for his psychic powers (Maha Moggallana) was said to be Mara in a past life (MN 50). In original Buddhism, psychic powers were not considered a sign of sanctity.
As one of the other answers mentioned I never heard of such a thing of monks saving people from disaster. Actually it would be a waste of time from buddhist point of view that instead of teaching and practicing dhamma, monks try to save people from an ever changing world. Buddha himself many a times when faced with such disasters reminded people to remember Anicca and meditate on it.
But talking about supernatural power in general, actually there had been mention of divine eye, divine ear, flying, walking over water, etc. in some discourse and it's said that every disciple experience this in the process. like in "DN 12 PTS: D i 224-Lohicca Sutta: To Lohicca"
more suttas in here.