From Vakil's foundations of algebraic geometry, page 158, 5.3.2"
We say such a property is affine-local: a property is affine-local if we can check it on any affine cover. (For example, reducedness is an affine-local property. Do you understand why?) Note that if U is an open subscheme of X, then U inherits any affine-local property of X. Note also that any property that is stalk-local (a scheme has property P if and only if all its stalks have property Q) is necessarily affine-local (a scheme has property P if and only if all of its affine open sets have property R, where an affine scheme has property R if and only if all its stalks have property Q). But it is sometimes not so obvious what the right definition of Q is; see for example the discussion of normality in the next section.
He says stalk-local properties are necessarily affine local. I feel like the converse is not necessarily true but I can't come up with a counterexample. What is an affine local property that is not stalk-local?
I think locally noetherian is such a property. A scheme is locally noetherian if and only if all of its affine opens are noetherian. But there exists a non-noetherian ring $A$ such that $A_{\mathfrak{p}}$ is noetherian for all prime ideals $\mathfrak{p}$. An example is $\prod_{i=1}^{\infty} \mathbb{Z}/2\mathbb{Z}$.