The maester towers in the capital were where the royal court trained its new priests and scholars.
We say priests and scholars as if they were two separate things, but most times they were the same thing, just called differently depending on the circumstances they were employed.
Both of them studied the same religious text, but scholars were priests who then chose to further continue their study in the human domain- gaining knowledge in literature, philosophy, art, languages, mathematics, and history.
And thus as it could seen these priests were not the same as the priests in the local temples and shrines.
The latter was employed on an ad-hoc basis by the local head priest. They could be young boys the temple had taken in from the local area to be used as servants and assistants.
Over time, they would then gradually become part of the clergy, having mastered the various chants and rituals by repeatedly hearing and seeing them over many years.