We study the formation of nanoholes and quantum rings in GaAs and AlGaAs surfaces by local droplet etching (LDE) with Ga and In. The quantum rings are formed by the droplet etching process and surround the nanohole openings. Our data show that a low As pressure is essential for LDE and that process conditions with high Ga coverage yield formation of additional hillocks or large hills. With atomic force microscopy we establish that the amount of material removed from the nanoholes is equal to the amount of material stored in the quantum rings. Basing on the experimental observations, we propose a model of nanohole and quantum ring formation.