I was suggesting that this is less "mysterious" than it seems to sound, once you use a hierarchical model, where one random variable's outcome is used as the parameter for another variable.

Imagine one random variable p at the top center of a page, and then several variables under it, each connected by an arrow coming from p.

The random variables down there are the coin flips (each results in 0 or 1), and they are identically and independently distributed with a Bernoulli(p) distribution. And p itself is a random variable which has a distribution (expressing how likely each p is).

I'm just clarifying that all this fits into normal probability theory, there is no need to introduce any sort of "meta-probability theory".