It is seen in comments here that the diagonal subgroup of the finite group $G \times G$ is core-free maximal iff $G$ is a nonabelian simple group.
This gives examples of decomposable primitive permutation groups. Are there others examples?
Question: What's the classification of the decomposable primitive permutation groups?
Yes, the only decomposable finite primitive permutation groups are groups $S \times S$ with $S$ a nonabelian simple group, acting on the cosets of a diagonal subgroup.
A decomposable group has at least two minimal normal subgroups. According to the O'Nan-Scott Theorem, the only type of finite primitive permutation group $G$ with more than one minimal normal subgroups, has exactly two such, $M$ and $N$, with $M \cong N \cong S^k$ for some nonabelian simple group $S$ and $k \ge 1$.
In the primitive action, on a finite set $X$, $M$ and $N$ are both regular normal subgroups. But then $N$ is the centralizer of $N$ in ${\rm Sym}(X)$ and vice versa, and hence also in $G$. But if $G$ is decomposable, then $M$ and $N$ must be subgroups of the two direct factors, and so $G = M \times N$. Then $G$ is primitive only when $k=1$.