Description of ideal of intersection of two projective varieties

Question

I am new to algebraic geometry and have been thinking about an analogue of the identity $I(X_1 \cup X_2) = I(X_1) \cap I(X_2)$ for $X_1 \cap X_2$, where $X_1, X_2 \subset \mathbb A^n$ are say affine algebraic sets (base field algebraically closed and characteristic zero). Since $I(X_1) \cup I(X_2)$ is obviously not an ideal in general, I was thinking that a natural thing to try and relate $I(X_1 \cap X_2)$ with, would be the ideal sum $I(X_1) + I(X_2)$. Of course the latter is contained in the former since the sum of any two polynomials vanishing on all of $X_1$ and $X_2$ respectively, would vanish on all of $X_1 \cap X_2$, as the individual summands would. And in general it doesn't seem like the opposite inclusion is true, however I am yet to find an explicit counterexample. What I am more curious about is whether there is anything we can say about the relation between these two ideals, with some additional assumptions if necessary (say at least one/both are irreducible etc.). Is there any other (not necessarily simpler but hopefully elegant) description of the ideal $I(X_1 \cap X_2)$? What about analogues for projective algebraic sets, varieties etc. (where homogeneity also needs to be taken into account)? Thanks.