Two years ago /u/gorbachev asked this question about the oracle of Delphi. The user quotes Allen Dulles as writing:
What did make the oracle more of a secular institution at a later date, as we know today, was the fact that the priests apparently had networks of informants in all the Greek lands and were thus often better appraised of the state of things on earth than the people who came for consultation. Their intelligence was by no means of divine origin, although it was proffered as such.
I'd always thought oracles avoided inaccuracy by embracing vagueness, not by gathering intelligence! What evidence is there that oracles (and similar people) had far-flung secret contacts to keep them abreast of the state of the world?
I can't comment on the question more broadly, but at least people in the ancient world were convinced of the possibility of divine knowledge coming from rather mundane sources. In the third century AD Diogenes Laertius, citing the third century BC biographer Hermippos claimed that Pythagoras received his apparently divine foreknowledge by hiding out in his mother's basement (Diogenes Laertius 8.41; trans. Daniel Ogden, Magic, Witchcraft and Ghosts in the Greek and Roman Worlds (Oxford: Oxford University Press, 2002, 10)):
"And Hermippus has something else to say about Pythagoras. For he relates that when he was in Italy he made a little chamber under the ground and told his mother to write down what happened on a tablet, indicating the time at which things took place, and to send them down to him until he came up again. This his mother did. After a time Pythagoras came up again emaciated and skeletal. He went into the assembly and claimed that he had come from Hades, and he read out to the people what had happened. They were beguiled by his words and wept and wailed. They believed that he was divine, and even handed their wives over to him, thinking that they would learn something from him. They became known as the Pythagoricae. This is what Hermippus says."