What do the derived functors of the symmetric powers look like? I understand that this is related to the homology of the symmetric groups, but I don't know a reference for that.

Namely, I'm interested in the homotopy groups of the free simplicial *commutative* ring on a simplicial set. Let $X_\bullet$ be a simplicial set; I'd like to know the homotopy groups of $\mathbb{Z}[X_\bullet]$. This is the symmetric algebra on the free simplicial abelian group $\mathbb{Z}X_\bullet$, which is weakly equivalent to a product of Eilenberg-MacLane simplicial abelian groups corresponding to the homology of $X_\bullet$ (and is cofibrant).
In particular, we have a weak equivalence of simplicial commutative rings
$$\mathbb{Z}[X_\bullet] \simeq \bigotimes \mathbb{L} \mathrm{Sym}^\bullet K( H_n(X_\bullet, n)),$$
which brings up the question of what $\mathbb{L} \mathrm{Sym}^\bullet$ looks like.
Tyler Lawson points out in answering this question that the answer is somewhat complicated and describes it in low degrees.

Is a complete answer known?