For a finite group acting linearly on a vector space, a separating set is a subset of the invariant ring that separates the orbits. In this paper, we determined explicit separating sets in the corresponding rings of invariants for four families of finite dimensional representations of the elementary abelian $p$-groups $(\mathbb{Z}/p{)}^2$ of rank two over an algebraically closed field of characteristic $p$, where $p$ is an odd prime. Our construction was recursive. The separating sets consisted only of transfers and norms, and the size of every separating set depended only on the dimension of the representation.