The cult followed the prophecies set down in the Sibylline Oracles by the original Sibyl herself. A Sibylline Sister could be recognised by stylised makeup which included eyes painted on the back of both hands.
In 62, an earthquake in Pompeii awoke the dormant Pyrovile who took over the Sisterhood and other cults. The members breathed in dust from the volcano which expanded their latent psychic powers but also slowly converted them into stone Pyrovile. The power to communicate telepathically came from the Pyrovile influence but the power of foresight was caused by the temporal rift created by the power of the eruption of Mount Vesuvius, echoing back through the alternative timeline in which the volcano did not erupt.
Sibylline Sisters in 79 included their High Priestess, Spurrina, Thalina, Evelina, and one other soothsayer. Evelina alone survived the eruption as the Tenth Doctor rescued her, with her family in the Doctor's TARDIS. (TV: The Fires of Pompeii)