Stone duality, one of many dualities between certain lattices and certain topological spaces, asserts that there is a contravariant categorical equivalence between the category $\text{Bool}$ of boolean algebras and the category $\text{Stone}$ of stone spaces. For those who are not familiar with this, here is a brief statement of what this says:
Definition: A boolean algebra is a partially ordered set $A$ such that
1) $A$ has finite meets and finite joins, including the empty join.
2) Meets distribute over joins in $A$
3) For each $a \in A$, there is $b$ such that $a \vee b$ is the greatest element, and $a \wedge b$ is the least element.
(1) says that $A$ is a bounded lattice, (1) and (2) say that $A$ is a bounded distributive lattice, and (3) says that $A$ is complemented.
Definition: A stone space $X$ is a space occuring as the cofiltered limit $\text{limit}\ X_i$ of discrete spaces in the category of topological spaces.
One functor $\text{Spec} : \text{Bool} \rightarrow \text{Stone}$ sends a boolean algebra $A$ to $\text{Bool}(A, \{ 0, 1 \})$ of maps of boolean algebras from $A$ to the boolean algebra $\{ 0, 1\}$. The other sends a stone space $X$ to $\text{Stone}(X, \{ 0, 1\})$, the set of maps of stone spaces from $X$ to $\{ 0, 1\}$.
I am interested in how this might work for $\sigma$-algebras. A $\sigma$-algebra is a subalgebra of the boolean algebra $P(X)$ (power set of a set) closed under countable meets (and therefore countable joins). By the categorical equivalence, $\sigma$-algebras $A$ on $X$ correspond to certain quotient objects of $\hat{X}$, the profinite completion of $X$ (inverse limit over all quotients onto a finite set).
Question: Let $X$ be a set. Can we characterize the quotient stone spaces of $\hat{X}$ corresponding to $\sigma$-algebras in $P(X)$?
Note: it was mentioned in the comments that $\sigma$-algebra can also refer to an ambient boolean algebra which has countable meets (and therefore countable joins). Here I mean to specifically fix an embedding into $P(X)$, and for joins and meets in the $\sigma$-algebr to match joins and meets in $P(X)$.