For simplicity I'm considering only the sphere case.
In the Gauss' Law formulation we have some field $E$ introduced by charges $Q$ inside some sphere, then we compute flux and integrate, and we get result $Q/e_0$. Right. But this formulation doesn't take into account any possible outer charges, because $E$ used in this law comes only from $Q$. I suppose that any outer field effects will cancel out, because flux coming from it must go through surfaces with opposite normals. But we didn't make any proof of that, as far as I have seen in Gauss' Law proofs.
So, when we measure electric field inside the sphere with uniformly distributed $Q$, we make some assumptions on uniformity of $E$ inside this sphere and we consider hollow Gauss' sphere with radius smaller than radius of the outer sphere. And then we use $E$ to compute total flux going through inner sphere and conclude that it must be zero. Right.
But why can we use this $E$? We don't know that there aren't any effects from outer charges fields. So why can we use Gauss' Law?