I don't know the answer, although I've thought about this quite a bit. I think the best that can be said in general is that the free cartesian monoidal category on $\mathcal{C}^\otimes$ is a full subcategory of $\text{Fun}^\otimes(\mathcal{C}^\otimes,\text{Set}^\times)^\text{op}$, which is the category of symmetric monoidal functors. That is Theorem 3.10 of https://arxiv.org/abs/1606.05606.

There is some hope of giving a more explicit description if we know something about maps *into* tensor products. Here are three examples. (I would love to understand what they have in common.)

**Baby example**: If $\mathcal{C}^\times$ is already cartesian monoidal, then $\mathcal{C}_{/Y\times Z}\cong\mathcal{C}_{/X}\times_\mathcal{C}\mathcal{C}_{/Y}$. That is, a map $X\rightarrow Y\times Z$ corresponds to two maps $X\rightarrow Y$ and $X\rightarrow Z$. In this case, the free cartesian monoidal category on $\mathcal{C}$ is $\mathcal{C}$ itself.

**Example 2**: If $\mathcal{C}^\amalg$ is cocartesian monoidal, we may ask whether the coproduct behaves like a disjoint union (that is, $\mathcal{C}$ is *disjunctive*) in the following sense: any map $X\rightarrow Y\amalg Z$ decomposes canonically as the coproduct of two maps $X_Y\rightarrow Y$ and $X_Z\rightarrow Z$. That is, $\mathcal{C}_{/Y\amalg Z}\cong\mathcal{C}_{/Y}\times\mathcal{C}_{/Z}$. In this case, I believe that the free cartesian monoidal category can be described as a category of spans, something like the following:

Objects are just objects of $\mathcal{C}$. Morphisms from $X$ to $Y$ are spans $X\leftarrow T\rightarrow Y$, such that $T$ admits a decomposition $T_1\amalg\cdots\amalg T_n$ with each $T_i\rightarrow X$ an inclusion of a direct summand; that is, $T_i\amalg T_i^\prime\cong X$.

For example, when $\mathcal{C}=\text{Fin}$, the category of finite sets, the free cartesian monoidal category is spans of finite sets. The $\mathcal{C}=\text{Fin}$ case is a main result of the paper cited above (at the level of $\infty$-categories). The general case is not written down anywhere, as far as I know.

**Example 3**: Given an operad $\mathcal{O}$, there is a category $[\mathcal{O}^\otimes]$, whose objects are finite sets. A map from $S$ to $T$ is a function $f:S\rightarrow T$ and an $|f^{-1}(t)|$-ary operation for each $t\in T$. Concatenation gives $[\mathcal{O}^\otimes]$ a symmetric monoidal structure.

Note that $[\mathcal{O}^\otimes]$ is disjunctive (in the sense of Example 2) but not cocartesian monoidal. In this case, the free cartesian monoidal category is the full subcategory of $$\text{Fun}^\otimes([\mathcal{O}^\otimes],\text{Set}^\times)^\text{op}=\text{Alg}_\mathcal{O}(\text{Set})^\text{op}$$ spanned by the free $\mathcal{O}$-algebras. Classically, the free $\mathcal{O}$-algebra on $n$ generators is $$\coprod_{k\geq 0}\mathcal{O}(k)\times_{\Sigma_k}n^k.$$ That effectively computes the free cartesian monoidal category on $[\mathcal{O}^\otimes]$ as a type of span construction, similar to Example 2. See the paper above, Remark 5.1 (also Example 3.16). The span construction can be made precise without much difficulty for ordinary categories, but it is an open problem for $\infty$-categories.