For simplicity this is about polynomials in just two variables.
Any $f\in\mathbb Q[X,Y]$ can be written as a linear combination of monomials $X^iY^j$ and therefore as a sum of polynomials $p_{ij}\in\mathbb Q[X^iY^j]$ over one variable:
$$\displaystyle f(X,Y)=p_{00}+\sum_{\gcd(i,j)=1}p_{ij}(X^iY^j).$$
My question: is the subring of all integer valued polynomials $f$ over two variables identical with the subring of all sums of integer valued polynomials $p_{ij}$ over one variable as above?
An integer valued polynomial in one variable is a polynomial $p$ with rational coefficients such that $p(\mathbb Z)\subseteq \mathbb Z$. And corresponding for polynomials over several variables.
It might be some abuse of language to call $p_{ij}$ polynomials over one variable, they may rather be polynomials over one variable applied to monomials $X^iY^j$.
The answer is that there is a counter-example is $\frac{X(X-1)Y(Y-1)}{4}$, as a user of Mathematics Stack Overflow found. The sums doesn't form a ring, just a group.
If I have understood your structure correctly, each $p_{ij}$ is a polynomial in one variable, but the variable is different for each combination of $i$ and $j$. So $p_{10}$ is a polynomial in $X$, $p_{01}$ is a polynomial in $Y$, $p_{11}$ is a polynomial in $XY$ etc. When you add the $p_{ij}$ terms together you have to keep these variables distinct. For example, $p_{10}(X) + p_{01}(Y) + P_{11}(XY)$ is a function of $X$, $Y$ and $XY$, not just of one "generic" variable.
Another complication is that each of the $p_{ij}$ can have a constant term, so the decomposition is not unique. Do you regard $X+Y+1$ as $(X) + (Y+1)$, or as $(X+1)+(Y)$ or even as $(X) + (Y) + (1)$ where $(1)$ is a (constant) polynomial in $XY$ etc.