I don't know what you are defining as "the middle and late childhood period", which makes giving specific conditions difficult - they vary at different ages.
Toddlers acquire language as they hear and imitate and receive reinforcement for what they have done. In this sense, necessary conditions are adequate hearing (language development for hearing-impaired individuals is much more difficult), role models in the sense of people using language around them for the children to hear frequently and clearly, and reactions to shape their use of language - when a toddler says something and receives a positive reaction to the noises s/he made, the child will remember and use that set of sounds again in a similar situation.
As children age, development of language becomes a process of refinement and expansion. Role models continue to play a critical part in providing patterns for word use and meaning, pronunciation, localized slang expressions, etc. Direct instruction may also be introduced to teach specifics of grammar or to provide direct definitions for new vocabulary. Mental and emotional factors also continue to be important - autistic children may have impaired language development because they are unable to make the connections to others necessary for language development to occur.
Middle childhood age would be ages 7–9
Late childhood age would be ages 10–11