The society is still fairly scattered, non-centralized, with individual villages striving for themselves. There might be some degree of trade, some degree of political centralization, but for the most part each village is on their own.
The primary right-performers are a priesthood. It's not an organized group, but women who are taught by the woman who had the job before her, who learned from the woman before her, etc, on back down since the gods only know when. Whenever the old priestess dies, her trainee takes the position, and the next girl born becomes the new trainee. There might be minor regional variations, of course, but that's pretty much the standard.
The priestess lives on the edge of town. A romantic entanglement or two in their teenage years is expected, perhaps even encouraged, provided that it ends before the girl become a priestess. If the girl becomes pregnant, the child is sent off to another village, so that the priestess can remain truly neutral and help her town as a group, rather than being distracted by any sort of protectiveness towards one individual, be it lover or child.
The rights which the priestess performs take a personal toll on her. Her life is sacrificed for the village. If she wants rain to come, she has to give her own blood. If she heals a broken bone, her own bones weaken a little. If she cures an illness, she'll cough a little bit more at night when the cold comes. That, combined with years of training, teach the priestess to use her magics as little as possible; for instance when asking for rain in a drought, she knows that the whole world is in drought, and that she understands the plight of the people around her, but if the rain gods could spare just a little water, it would make a huge difference. She might not produce a bumper crop, but she'll keep things alive. It's personally painful to her, its slow, but its balanced.
Into this world a hierarchical priesthood comes. They are mostly young men who have figured out how to make rites that work and are young and idealistic enough to have no idea why they wouldn't use their power however they can. They do it without much personal cost to themselves, too. Their trick is that they hold group rituals and help direct the will of the people. The people, not being trained in such things, tend to shout rather than whisper. The gods might not be able to ignore their cries for rain, but might be sufficiently irritated that they'll send a flood. It may be bad, but it's damn impressive.
This hierarchy sends out missionaries. One of these missionaries find themselves a village that he believes to be vulnerable somehow (youngish priestess who hasn't quite gelled into her position yet, or an old woman who isn't going to have this fight, and believes that the world will return to her way sooner or later, so why have a big confrontation?) This man begins by claiming that his book is not the only source of truth, but it is a powerful one, so why not pray to both? (Jesus will save your soul, but Thor will save your boat kind of thing.) When he holds rights, he'll have the whole village there, but make certain that the priestess has a place of honor, that it's them doing the rites together, but she can also do her own rites on her own.
Until, of course, things goes wrong. The priest has the advantage here: he can say that the people are at fault, that it's their belief that faltered, so clearly not his fault. The priestess knows damn well that it was all her, and can't help but wonder if it's not her fault, if she felt too much the drain on her blood when she asked for rain, if she feared her aching bones, or her faltering lungs, and looked away from the gods for just a minute, and now they've abandoned her.
When the people of the village are looking for someone to blame for their issues, they'll go for her. Because the priest at least clearly did something, immediate and flashy and maybe too extreme but hey, that's what happens. And she could have done more, but didn't. She wasn't willing to make that sacrifice.
So, with a little encouragement from their priest, they sacrifice her.
They feel awful about it later, though. The church feels their pain, and sends out a replacement for the priest, someone less extreme. They'll be sure to roundly chastise the over-zealous priest, and make sure he's dealt with properly.
Which means, of course, that they'll send him back out to another village to find some more converts.
Of course, it wouldn't work everywhere. That's what Katakaw is for.