Thank you SO MUCH for this amazing answer. Now I see why my naive approach did not succeed, more machinery is needed here. I’ll take a careful look at all this concepts. I really appreciate your explanation.

@JasonStarr ... of the question I posted? It's just because in the graded case it can be proved in a rather elementary way, and I'd like to explore first this possibility before getting into all this sheaf-theoretic machinery you exposed. Again, thank you very much for time, your comments are really appreciated.

@JasonStarr Thank you very much for the explanation and the references. I'll take a careful look at all this concepts because there are some of them I'm totally unfamiliar with. The question I posted arose when I was studying the concept of regularity of sheaves and the possibility of using it to define the Hilbert scheme for products of projective spaces from a 'bigraded point of view', in a similar way we do for the Hilbert scheme in the usual projective space. Anyway, I'd like to introduce all this stuff in a more elementary way, so, do you know if there is a purely algebraic proof...

Thanks! My first approach was using Hilbert polynomial, but I’m not sure how to get rid of the undesirable embedded components that may appear when treating with the sum of the ideals, just to get nice exact sequences that give you the degree of the intersection (counting multiplicity)