Is language distinct from other cognition during development? Does neural machinery for language emerge from general-purpose neural mechanisms, becoming tuned for language after years of experience and maturation? Answering these questions will shed light on the origins of domain-specificity in the brain. We address these questions using precision fMRI, scanning young children (35 months to 9 years of age) on an auditory language localizer, spatial working memory localizer (engaging the domain-general multiple demand [MD] network), and a resting-state scan. We create subject-specific functional regions of interest for each network and examine their selectivity, specificity, and functional connectivity. We find young children show domain-specific, left-lateralized language activation, and that the language network is not responsive to domain-general cognitive load. Additionally, the cortically adjacent MD network is selective to cognitive load, but not to language. These networks show higher within versus between-network functional connectivity. This connectivity is stable across ages (examined cross-sectionally and longitudinally), whereas language responses increase with age and across time within subject, reflecting a domain-specific developmental change. Overall, we provide evidence for a double dissociation of the language and MD network throughout development, in both their function and connectivity. These findings suggest that domain-specificity, even for uniquely human cognition like language, develops early and distinctly from mechanisms that presumably support other human cognition.