The flat locus is open in any event under reasonable hypotheses (EGA IV-3, Th. 11.1.1), so being flat "at a point" and "in a neighborhood of a point" are equivalent.
Also, this particular result is true more generally: if $f: X \to Y$ is a morphism of finite type $S$-schemes ($S$ being noetherian), and if $X, Y$ are flat, then $f$ is flat if and only if each of the maps $f_s: X_s \to Y_s$ are flat. This follows from the fact that one direction of the local criterion of flatness is true under more generality: that is, if $B \to C$ is a morphism of local noetherian rings both local and flat over the local noetherian ring $(A, \mathfrak{m})$, then $B \to C$ is flat if and only if the fiber $B/\mathfrak{m} B \to C/\mathfrak{m}C$ is flat. See for instance Proposition 4.10 of this document.