Why are finite intersection of affine open sets of a scheme not always affine?

Question

I seem to have confused myself about the notion of separatedness. I am not even sure how it makes sense to say that the intersection of affine open subsets is affine when a scheme is separated, since separatedness is a property of morphisms and not of schemes. The obvious response to this is that when we talk about a scheme being separated, we mean that the structure morphism to $\operatorname{spec}\mathbb{Z}$ is separated. But take any scheme $X$, with affine open sets $\operatorname{spec}A$ and $\operatorname{spec}B$. Then both of these open subsets are schemes over $\operatorname{spec}\mathbb{Z}$ by virtue of being schemes. But then why isn't the intersection of $\operatorname{spec}A$ and $\operatorname{spec}B$ just given by $\operatorname{spec}(A \otimes_{\mathbb{Z}} B)$? The scheme $\operatorname{spec}A$ and $\operatorname{spec}B$ are the same whether we view them as open subsets of $X$ or not. So why should their intersection be dependent on that? I'm sorry if this is a silly question but I'm completely confused.