There are homology isomorphisms $K(Br,1)\to \Omega^2_0 S^2$ and $\Omega^2 S^3\to \Omega^2_0 S^2$, so you are really asking about the homology of the stable braid group $Br$ (the colimit of the natural inclusions $Br_n\hookrightarrow Br_{n+1}$).

As expected there is no neat description with integral coefficients, but much is known. You'll find a nice summary in Section 4 of this paper of Vershinin .

There are homology isomorphisms $K(Br,1)\to \Omega^2_0 S^2$ and $\Omega^2 S^3\to \Omega^2_0 S^2$, so you are really asking about the homology of the stable braid group $Br$ (the colimit of the natural inclusions $Br_n\hookrightarrow Br_{n+1}$).

As expected there is no neat description with integral coefficients, but much is known. You'll find a nice summary in Section 4 of this paper of Vershinin .