Separating path systems in complete graphs

We prove that in any $n$-vertex complete graph there is a collection $\mathcal{P}$ of $(1+o(1))n$ paths that strongly separates any pair of distinct edges $e,f$, meaning that there is a path in $\mathcal{P}$ which contains $e$ but not $f$. Furthermore, for certain classes of $n$-vertex $\alpha n$-regular graphs we find a collection of $(\sqrt{3\alpha +1}-1+o(1))n$ paths that strongly separates any pair of edges. Both results are best-possible up to the $o(1)$ term.