By "separated", I mean that the morphism $\mathrm{Proj\,}S\to\mathrm{Spec\,}\mathbb{Z}$ is separated. Basic commutative algebra tells us that that the morphism $\mathrm{Spec}\,B\to\mathrm{Spec}\,A$ is always separated, but those methods don't seem to work when dealing with $\mathrm{Proj}$.
My goal is really to prove that the morphism $\mathbb{P}_\mathbb{Z}^n\to\mathrm{Spec}\,\mathbb{Z}$ is separated, which I know must be true, though I don't know how to prove it.
Look at prop 3.6 of page 100 of Liu. A scheme is separated iff it is covered by affine $U_i$ such that $U_i\cap U_j$ is also affine and$$\mathcal{O}(U_i)\otimes \mathcal{O}(U_j)\to \mathcal{O}(U_i\cap U_j)$$ is surjective.
The standard covering $D_+(x_i)$ works for $\mathbb{P}^n$.
It only remains to see that the map $$k\left[\frac{x_1}{x_0},\frac{x_2}{x_0}\right]\otimes k\left[\frac{x_0}{x_1},\frac{x_2}{x_1}\right]\to k[x_0,x_1,x_2]_{x_0x_1}$$ is surjective and this is clear.
Indeed this generalizes to any graded ring $S$. We need to see that $$S_{(f)}\otimes S_{(g)}\to S_{(fg)}$$ is surjective and if $f$ has degree $k$ and $g$, degree $l$, then we have
$$\frac{a}{f^ng^m}=\frac{ag^{(k-1)m}}{f^{n+lm}}\frac{f^{lm}}{g^{km}}$$