In nature, many skilled behaviors are learned without explicit reinforcement. Instead, this learning is thought to be guided by an internal estimate of performance error, yet the mechanisms by which such estimates are acquired remain unknown. In male zebra finches, juveniles are first tutored by an adult male during a sensory learning phase, followed by a sensorimotor phase in which they practice by themselves in order to replicate the song. Inspired by models of corollary discharge and predictive coding, we hypothesize that in the sensory phase, juvenile auditory areas gradually learn to cancel the predicted auditory inputs of the tutor song. Later, during the sensorimotor phase, these circuits produce error codes when the bird's current performance fails to match the tutor song. To investigate this hypothesis, we built models in which a local circuit receiving both premotor and auditory input learns to cancel the auditory patterns of the tutor song via anti-Hebbian plasticity. Testing with real auditory data, we found that, while different circuit motifs and sites of synaptic plasticity can learn to produce error codes, balanced E-I networks with effectively anti-Hebbian plasticity in the recurrent connections agree better with existing experimental observations than the other motifs or those with plasticity in the premotor projections. We further found that the recurrent connections learn a distinct ``landscape'' mode, which shapes the sensitivity of error responses, and many ``memory'' modes, which encode implicit memory of the tutor song in a distributed manner. Together, our results suggest that purely local learning mechanisms for predictive cancellation are sufficient for computing multidimensional error signals capable of guiding self-directed learning of natural behaviors.