'local' originates from Latin, specifically the word 'localis' (from 'locus'), where 'locus' meant 'place'; 'language' originates from Old French 'language', ultimately from Latin 'lingua', where 'lingua' meant 'tongue'.

'local' came into English via Latin 'localis' and Medieval Latin, becoming Middle English 'local' and then modern English 'local'. 'language' changed from Old French 'language' (from Latin 'lingua') into Middle English 'language' and eventually modern English 'language'.

Initially, 'local' simply meant 'of a place' and this sense has remained; 'language' originally meant 'tongue' (both the organ and, by extension, speech) and evolved into the broader modern meaning 'system of communication by speech'.