A reducible scheme is singular along the intersection of two components: this is usually proved using the well-known, but not entirely trivial, fact that a regular local ring is an integral domain. Another way of saying this is that, as sets, the intersection of the components is contained in the singular locus. I'm wondering to what extent this is true as schemes.

Let $X = X_1 \cup X_2$ be the union of two affine varieties in $\mathbb{A}^n$, which maybe we'll assume to have the same dimension $d$. Let $S \subset X$ be the closed subscheme defined by the Jacobian criterion, that is, the subscheme cut out by the determinants of the $(n-d)\times(n-d)$ minors of the Jacobian matrix of some set of defining polynomials for $X$. Is $X_1 \cap X_2$ scheme-theoretically contained in $S$? If so, is there a reasonably elementary way to see this?