Their power comes from their worshipers, the more they have, and the more devout they are, the more powerful the pantheon, though gods with fewer worshipers than men in a moderately large city are so weak they are irrelevant for all practicable purposes.
But in order to act they need a bridge between themselves and the mortal world, a mortal (the Singers existence proves they do not need to be human) who has awakened the magic in their blood and is sworn to them.
Rather embarrassingly, this was, while not common knowledge, still well known among the upper ranks of the Essosi religions, to the point being a mage-priest was a de-facto requirement for attaining high office in many of them (the fire priests of R'hllor are a good example).
The Old Gods, though the Green Men, also informed that while many deities and pantheons liked to proclaim themselves all-knowing, the truth was that their ability to gather information, and generally act on the world, even with a mage-priest, sharply decreases the further they get from their domain. For the Old Gods, their domain is wherever the weirwoods grow, with godswoods and heart trees being local nexuses of their power that magically inclined Green Men can tap into.
…The Seven-who-are-One do not have any mage-priests among their Septons or Septas in this age, but they did once. When the Andals first invaded they brought mage-Septons with them, who managed to successfully counter the magically inclined Green Men and Singers who faced them due to having blooded themselves against the far more dangerous mages of Valyria…
…Even back then the Andals and the Seven had a deep antipathy towards magic, likely a reactionary movement from dealing with the horrifyingly creative nastiness the Valyrian Freehold's mages came up with, and after they conquered all the kingdoms south of the Neck that antipathy turned to persecution.
The fact that killing everyone with magic meant that there wouldn't be any who could become mage-priests doesn't seem to have occurred to the Seven, or if they did eventually realize it was at far too late a date to stop it.
The last mage-Septon that can be definitively identified died in 849BC, burnt alive on charges of witchcraft by a High Septon who in turn would be executed for a long list heinous crimes…
…Even ignoring the obvious aspects, the Green Men differ greatly from Septons. For starters, while they do forswear any prior allegiances and inheritances, they do not take a vow of chastity and are allowed to marry and have a family, those with magic in their blood being greatly encouraged, both societally and divinely, to have the latter. As the Old Gods are deities of nature denying those in their service the opportunity to reproduce runs counter to their nature. They also allow women to join…
…There was a great deal of concern when Baelor the Oathbreaker ascended the throne as many worried that he might awaken as the first mage-Septon of the Seven since before the Doom of Valyria. Especially when one of his earliest acts was to have the Red Keep's godswood cut down.
Eventually the conclusion was reached that he was not a mage-Septon, namely because none could believe that the Seven would let their mage-priest be so incompetent if they were whispering in his ear, rather than the complete dearth of miracles the King routinely failed to perform.
Some have humorously suggested that the Seven were whispering in Baelor's ear, but he was just that inept, and would have been far worse without their guidance. Others have posited that the king could hear them, but with great difficulty, like trying to listen to someone shouting to you from a great distance.
All the same, Baelor's unthinking destruction of the ancient treaties not only between the Targaryens and the North, but between the Andals and the First Men, very nearly tore the realm apart into a religious war between the Seven and the resurgent Old Gods.
Mobs of zelous smallfolk and hedge knights led by ambitious and/or bigoted Septons ravaged the Riverlands, burning and cutting down any weirwood or heart tree they could find and lynching any who tried to stop them. Of course the First Men did not take this lying down and gave as good as they got, especially after the North, with Lord Stark willfully turning a blind eye, sent men and arms, to assist in response to Baelor forbidding Lord Tully from calling his banners to put down the unrest…
…The Clash of the Faiths ended with the death of Baelor, as the moment Lord Tully received word of the King's death he called his banners and rode out to put his kingdom back to order. The First Men and Northern soldiers largely stood down peacefully, but the Faith, who were rapidly reforming into a new Faith Militant, did not, resulting in several bloody clashes before the order was put down once more.
In the end the Old Gods won the Clash, successfully converting over half of the Riverlands' smallfolk to their faith, thanks to the Green Men performing miracle after miracle in front of the smallfolk (which the Maesters and Septons wasted no time in discrediting and "debunking" but only convinced the nobility, who remained largely uninvolved, and the other southern kingdoms) and re-settled High Heart, the weirwood grove resurrected with new growth from the stumps through the ritual sacrifice of thirty-one septons who took up arms against them, one per stump.
That said, large and strong bastions of the Seven remain in the Riverlands, mainly around the major settlements like the Twins, Seagard, Riverrun, etc. which saw no fighting or attempted conversions, none wanting to give Lord Tully an excuse to disregard King Baelor's command…
…In a reversal of historical trends, since the Clash, the Faiths of the Old Gods have been slowly expanding, spreading into the Westerlands and crownlands, something that is a constant source of strife as septons still clash with Green Men…