We describe an effective procedure to compute the local subgroup zeta functions of the free class-$2$-nilpotent groups on $d$ generators, for all $d$. For $d=4$, this yields a new, explicit formula. For $d\in\{4,5\}$, we compute the topological subgroup zeta function. We also obtain general results about the reduced and topological subalgebra zeta functions. For the former, we determine the behaviour at one and for the latter, the degree and behaviours at zero and infinity. Some of these results confirm, in the relevant special cases, general conjectures by T. Rossmann.