Locally Free Sheaves

As the name indicates, "locally free" is a local concept.
So we can assume that $X=Spec(A)$, the affine scheme associated to the noetherian ring $A$, and $F=\tilde M$, the coherent sheaf associated to the finitely generated $A$-module $M$.

The sheaf $F=\tilde M$ is locally free if and only if the module $M$ is projective.
And a finitely generated module $M$ over a noetherian ring $A$ is projective if and only if all its localizations $M_{\mathfrak p} \; ( \mathfrak p \in Spec(A)$ are $A_{\mathfrak p}$-free modules.
Since at a point $p\in X$ corresponding to the prime $\mathfrak p \in Spec(A)$ we have $\mathcal O_{X,x}=A_{\mathfrak p}$ and $ F_p=M_{\mathfrak p}$, we see that indeed freeness of all stalks of $F$ implies local freeness of the given coherent sheaf $F$.