A possibly interesting analogue of the formula $HF_{2*} HF_2 = \otimes_{i\ge1} F_2[\xi_i]$ is $HZ_{(2)*} HZ_{(2)} = \otimes^L_{i\ge1} Z_{(2)*}[\xi_i^2]/(2\xi_i^2)$, where $\otimes^L$ means the derived tensor product. In other words, resolve $Z_{(2)*}[\xi_i^2]/(2\xi_i^2)$ by (flat or) free $Z_{(2)}$-modules, tensor the resolutions together, and pass to homology. If I recall correctly, the "first" interesting class $\xi_2^3 + \xi_1^2 \xi_3$ (in degree 9) arises as a torsion product of $\xi_1^2$ and $\xi_2^2$. I needed this for a Shukla homology calculation once. Presumably there is also an odd story.

A possibly interesting analogue of the formula $H\mathbb{F}_{2*} H\mathbb{F}_2 = \otimes_{i\ge1} \mathbb{F}_2[\xi_i]$ is $H\mathbb{Z}_{(2)*} H\mathbb{Z}_{(2)} = \bigotimes^\mathbb{L}_{i\ge1} \mathbb{Z}_{(2)*}[\xi_i^2]/(2\xi_i^2)$, where $\otimes^{\mathbb{L}}$ means the derived tensor product. In other words, resolve $\mathbb{Z}_{(2)*}[\xi_i^2]/(2\xi_i^2)$ by (flat or) free $\mathbb{Z}_{(2)}$-modules, tensor the resolutions together, and pass to homology. If I recall correctly, the "first" interesting class $\xi_2^3 + \xi_1^2 \xi_3$ (in degree 9) arises as a torsion product of $\xi_1^2$ and $\xi_2^2$. I needed this for a Shukla homology calculation once. Presumably there is also an odd story.