Let $X$ be a topological space, and $\mathcal{F}_1$, $\mathcal{F}_2$ and $\mathcal{F}_3$ be sheaves on $X$. Suppose for all $U$ open in $X$ we have,
$0\longrightarrow \mathcal{F}_1(U)\longrightarrow\mathcal{F}_2(U)\longrightarrow\mathcal{F}_3(U)\longrightarrow 0$ is exact, then I think that it is not necessarily true that the sheaf sequence $0\longrightarrow\mathcal{F}_1\longrightarrow\mathcal{F}_2\longrightarrow\mathcal{F}_3\longrightarrow 0$ is exact.
But if we take direct limit over all open sets $U$ in the first sequence, since direct limit preserves exactness, we get $0\longrightarrow\mathcal{F}_{1_x}\longrightarrow\mathcal{F}_{2_x}\longrightarrow\mathcal{F}_{3_x}\longrightarrow 0$ is exact for all $x$.
That is we have stalk level exactness, therefore the sheave sequence is exact too. Is this right?