Let $X$ be an equidimensional reduced scheme of finite type over an algebraically closed field $k$. Assume that $X$ has two irreducible components $X_1$ and $X_2$. Assume also that $X_1$ and $X_2$ are nonsingular varieties which meet transversally at $Y := X_1 \cap X_2$, so that $Y$ is a nonsingular variety of codimension $1$ in $X_1$ and in $X_2$. If it helps, we may assume that $Y$ is again irreducible.
We may consider the two normal bundles $\mathcal N_{Y/X_1}$ and $\mathcal N_{Y/X_2}$, which are actually line bundles on $Y$. Thus, they define elements of $\mathrm{Pic}(Y)$. Are these two elements inverse of eachother? In other words, is the line bundle $\mathcal N_{Y/X_1} \otimes \mathcal N_{Y/X_2}$ isomorphic to $\mathcal O_Y$ ?
No in general. For instance, if $X_i$ are hypersurfaces of degree $d_i$ in the projective space $\mathbb{P}^n$ then $$ \mathcal{N}_{Y/X_1} \cong \mathcal{N}_{X_2/\mathbb{P}^n}\vert_Y \cong \mathcal{O}_Y(d_2) $$ and similarly $\mathcal{N}_{Y/X_2} \cong \mathcal{O}_Y(d_1)$.
If, however, $X = X_1 \cup X_2$ is a fiber of a flat morphism to a curve, then the relation that you want is indeed true.