Do children learn language from the words that they produce themselves? Because children know that they have imperfect knowledge of language, they could simply ignore their own productions. However, children could also learn from their productions -- using what they say and how their caregivers respond to update their language models. Using irregular plurals as a case study, we conducted a large-scale corpus analysis and two experimental studies to understand the role of children's productions and caregivers' responses in language learning. We demonstrate that children do learn from their own production, with errorful utterances leading to more errors. However, at least in some contexts, children can use implicit corrections from parents to offset the negative effects of their errors. Children thus appear to learn not only from their caregivers' productions, but also from their own productions and from the relationship between the two.