If $P_1, P_2$ are two random variables taking support in $[0,1]$ with correlation $\rho$,
$$ \operatorname{Corr}(P_1, P_2) = \rho $$
and $X_1 \sim \operatorname{Bern}(P_1)$, $X_2 \sim \operatorname{Bern}(P_2)$,
how can I find $\operatorname{Corr}(X_1, X_2)$?
I presume the $X_1 \sim \operatorname{Bern}(P_1)$ is intended to mean that the conditional distribution of $X_1$ given $P_1 = p_1$ is Bernoulli with parameter $p_1$, i.e. $\mathbb P(X_1 = 1 \mid P_1 = p_1) = p_1$ and $\mathbb P(X_1 = 0 \mid P_1 = p_1) = 1-p_1$, and similarly $\mathbb P(X_2 = 1 \mid P_2 = p_2) = p_2$ and $\mathbb P(X_2=0 \mid P_2 = p_2) = 1-p_2$. But that is not enough to tell us much about the dependence of $X_1$ and $X_2$ on each other. You need more assumptions. For example, perhaps $X_1$ is conditionally independent of $P_2$ and $X_2$ given $P_1$, and $X_2$ is conditionally independent of $P_1$ and $X_1$ given $P_2$. Then
$$ \eqalign{ \mathbb E [ X_1 X_2 ] &= \mathbb E \left[ \mathbb E [X_1 X_2 \mid P_1, P_2] \right]\cr &=\mathbb E \left[\mathbb E[X_1 \mid P_1, P_2]\; \mathbb E[X_2 \mid P_1, P_2]\right]\cr &= \mathbb E\left[\mathbb E[X_1 \mid P_1]\; \mathbb E[X_2 \mid P_2]\right]\cr &= \mathbb E[P_1 P_2]}$$ and then $X_1$ and $X_2$ have the same covariance as $P_1$ and $P_2$.