Let $G$ be a locally compact unimodular group, let $1\leq p<\infty$, let $\phi\in L^\infty(G)$ and assume that the Fourier multiplier $M_\phi$ associated with $\phi$ is bounded on the noncommutative $L^p$-space $L^p(VN(G))$. Then $M_\phi\colon L^p(VN(G))\to L^p(VN(G))$ is separating (that is, $\{a^*b=ab^*=0\}\Rightarrow\{M_\phi(a)^* M_\phi(b)=M_\phi(a)M_\phi(b)^*=0\}$ for any $a,b\in L^p(VN(G))$) if and only if there exists $c\in\mathbb C$ and a continuous character $\psi\colon G\to\mathbb C$ such that $\phi=c\psi$ locally almost everywhere. This provides a characterization of isometric Fourier multipliers on $L^p(VN(G))$, when $p\not=2$. Next, let $\Omega$ be a $\sigma$-finite measure space, let $\phi\in L^\infty(\Omega^2)$ and assume that the Schur multiplier associated with $\phi$ is bounded on the Schatten space $S^p(L^2(\Omega))$. We prove that this multiplier is separating if and only if there exist a constant $c\in\mathbb C$ and two unitaries $\alpha,\beta\in L^\infty(\Omega)$ such that $\phi(s,t) =c\, \alpha(s)\beta(t)$ a.e. on $\Omega^2.$ This provides a characterization of isometric Schur multipliers on $S^p(L^2(\Omega))$, when $p\not=2$.