What is the meaning of the notation $\theta \sim p(\theta)$ and $y \sim p(y|\theta)$? I've seen this notation pop up in some Bayesian statistics notes but despite my best efforts with Google I've not had any luck discovering the meaning. The only helpful context that I have is that $y$ is the observed data and $\theta$ is some sort of parameter. As for giving an example of this notation in use, I've seen De Finetti's theorem summarized as "$Y_1 , ... , Y_n | \theta$ are i.i.d. and $\theta \sim p(\theta)$ if and only if $Y_1 , ... , Y_n$ are exchangeable for all $n$".
EDIT: Note, I'm not asking how something like $Y \sim N(\mu, \sigma^2)$ is read, I know exactly what that means. It's the $p$ parts that are confusing me.
 A: The notation is overloaded (or abused) so $p$ refers both to the probability density function as well as to the distribution. This seems a lot simpler than to have say capital letters indicate the distribution and lower case letters indicate the pdf/pmf. 
Alternatively the $\sim$ notation can be thought of as overloaded so that it means the random variable on the left has the distribution on the right if the quantity on the right is a distribution. But if the quantity on the right is a pdf/pmf then $\sim$ means that the random variable on the left has the pdf/pmf of the quantity on the right.
A: The notation is very misleading with $p(\cdot)$ being heavily overloaded with multiple meanings and complete disregard for the difference between random variables and the values that they take on.
There is a parameter $\theta$ of unknown value which is modeled as a random variable $\Theta$, discrete or continuous depending on what $\theta$ is believed to be. Assuming that $\theta$ is believed to take on discrete values, $$\Theta \sim p_\Theta(\theta)\tag{1}$$ is saying that $\Theta$ is being modeled as a discrete random variable with probability mass function $p_\Theta(\theta)$, or to paint refined gold and gild the lily, the function $p_\Theta(\cdot)$ has the property that $$P\{\Theta = \theta_i\} = p_\Theta(\theta_i).$$ There are $n$ observations $x_1, x_2, \ldots, x_n$ which are also modeled as random variables $X_1$, $X_2$, $\ldots$, $X_n$. Assuming that everything is discrete, on a trial of the experiment, these random variables $\Theta, X_1, X_2, \ldots, X_n$ have values $\theta, x_1, x_2, \ldots, x_n$ of which we can observe only $x_1, x_2, \ldots, x_n$.  Now, the conditional probability mass function of the $X_i$ given that $\Theta$ has value $\theta$ is $$p_{X_1, X_2, \ldots, Xn \mid \Theta = \theta}(x_1, x_2, \ldots, x_n \mid \Theta = \theta)\tag{2}$$
meaning that 
$$P\big\{X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n \mid \Theta = \theta_i
\big\} = p_{X_1, X_2, \ldots, Xn \mid \Theta = \theta_i}(x_1, x_2, \ldots, x_n \mid \Theta = \theta_i).$$
Notice that $p$ has different subscripts in $(1)$ and $(2)$ to emphasize that there are two different functions being used here; one with just one argument and another with $n$ arguments, a far better arrangement than $p$ meaning different things depending on where it occurs and also better than the "canonical"

Let $X \sim f(x)$ and $Y \sim f(y)$ be random variables $\ldots$

or the physicists'

$X \sim f(X), Y \sim f(Y), X,Y \sim f(X,Y)$

where the density of $X$ can be quite different from the density of $Y$ and one is supposed to figure out from the letter used as the argument of $f\cdot)$ as to whether the density function of $x$ is meant or the density function of $Y$.
