We study dynamo processes in a convective layer of Boussinesq fluid rotating about the vertical. Irrespective of rotation, if the magnetic Reynolds number is large enough, the convection acts as an efficient small-scale dynamo with a growth time comparable with the turnover time and capable of generating a substantial amount of magnetic energy. When the rotation is important (large Taylor number) the characteristic horizontal scale of the convection decreases and the flow develops a well-defined distribution of kinetic helicity antisymmetric about the mid-plane. We find no convincing evidence of large-scale dynamo action associated with this helicity distribution. Even when the rotation is strong, the magnetic energy at large scales remains small, and comparable with that in the non-rotating case. By externally imposing a uniform field, we measure the average electromotive force. We find this quantity to be extremely strongly fluctuating, and are able to compute the associated $\alpha$-effect only after very long time averaging. In those cases for which reasonable convergence is achieved, the $\alpha$-effect is small, and controlled by the magnetic diffusivity. Thus we demonstrate the existence of a system whose small-scale dynamo growth rate is turbulent, i.e. independent of diffusivity, but whose $\alpha$-effect is laminar, i.e. dependent on diffusivity. The implications of these results to the problem of the generation of strong mean fields are discussed.