Diffusion of small two-dimensional Cu islands (containing up to 10 atoms) on Cu(111) has been studied using the newly developed self-learning Kinetic Monte Carlo (SLKMC) method which is based on a database of diffusion processes and their energetics accumulated automatically during the implementation of the SLKMC code. Results obtained from simulations in which atoms hop from one fcc hollow site to another are compared with those obtained from a parallel set of simulations in which the database is supplemented by processes revealed in complementary molecular dynamics simulations at 500 K. They include processes involving the hcp (stacking-fault) sites, which facilitate concerted motion of the islands (simultaneous motion of all atoms in the island). A significant difference in the scaling of the effective diffusion barriers with island size is observed in the two cases. In particular, the presence of concerted island motion leads to an almost linear increase in the effective diffusion barrier with size, while its absence accounts for strong size-dependent oscillations and anomalous behavior for trimers and heptamers. We also identify and discuss in detail the key microscopic processes responsible for the diffusion and examine the frequencies of their occurrence, as a function of island size and substrate temperature.