This is of course true, for any semi-separated scheme (i.e. the diagonal is affine), or maybe you assume $X$ is separated if you like. The reason that the sheaf is flat is that ${\rm Spec}(O_{X,x})\to X$ is affine and flat. In general if $f: X\to Y$ is flat and affine then $f_*O_X$ is $O_Y-$flat. This is obvious.
But I don't think it is true that $f: X\to Y$ is flat implies $f_*O_X$ is $O_Y-$flat. I would be happy if someone can back me up with a counter example here.