I think the oni are priests. The record stated that kidou is the foundation on which the kami are built. As we know from the stimulation game, the kami use worship as the source of their strength. So we can say that kidou is mantra in the form of reverence.
It's different from the other three hearts by both its colour (white instead of black) and size ( way bigger). The size is pretty easy to understand, worship increases with number, but the colour part is tricky. Kidou is white because like sennin, the oni becomes detatched from humans?
Chapter two mentions two definitions for oni: one is those who wouldn't be considered to be living properly and the other is man-eaters. Kidou is also said to be the art of achieving prosperity through the selfish act of harvesting people. So we can consider kidou to be living by extracting resources from other people instead of living properly and making stuff yourself.
The description of the three heart sword and koto-sama's story shows that the oni are necessary for bringing the kami to earth in the first place and that the oni's mantra are unusual.
With all of these in mind, my theory is the oni are proto-priests. Their mantras are unusual because unlike normal people who live properly and make a living for themselves, these people spend time writing story, fiction, illusions,... And through the stories of the priests the kami came to life. Priests, like the kami, are revered and given offering. So the source of their power is the same as the kami, they gain resources from other people, through the use of fiction, mantra and become powerful