Let $\mathcal F, \mathcal G$ be sheaves of $\mathcal O_X$-modules on a ringed space $(X,\mathcal O_X)$. I know that every morphism of $\mathcal O_X$-modules $\mathcal F\to \mathcal G$ induces a homomorphism of $\mathcal O_{X,x}$-modules $\mathcal F_x\to \mathcal G_x$ on every stalk.
Is a morphism $\mathcal F\to\mathcal G$ completely determined by these induced homomorphisms $\mathcal F_x\to \mathcal G_x$? Under what circumstances is this true?
This is always true, for all morphisms of sheaves. The module structure is irrelevant, but I will stick to it to use your notation.
Let $U \subseteq X$ be an open subset and $\phi: F \rightarrow G$ be a morphism of sheaves of $\mathcal{O} _{X}$-modules. Let $s \in F(U)$ be a section. Notice that $\phi(s) \in G(U)$ is completely determined by all its stalks by sheaf property, but $\phi(s) _{x} = \phi _{x} (s _{x})$. So $\phi$ is completely determined by all of $\phi _{x}: F_{x} \rightarrow G_{x}$.