We study the radiation dynamics of an electric dipole source placed near or inside a discrete invisibility cloak. We show that the main features of radiation dynamics can be understood in terms of the interaction of the source with a nonideal cloak in which local-field effects associated with the discrete geometry play a crucial role. As a result, radiation dynamics in a discrete cloak can differ drastically from what a source would experience in an ideal, continuous cloak. This can lead, for instance, to an enhancement of the emission by the cloak, thus making the source more visible to an outside observer than it would be without the cloak. The two main physical mechanisms for enhanced, or inhibited, radiation dynamics are the coupling of the source to leaky modes inside the cloak, and the coupling of the source with the lattice of the discrete cloak, via the local field. We also explore the robustness of the effect to material dispersion and losses