Watson-Crick finite automata are language recognizing devices similar to finite automata introduced in DNA computing area. Local languages are of great interest in the study of formal languages using factors of length two or more. We define Watson-Crick local languages using double stranded sequences where the two strands relate to each other through a complementary relation inspired by the DNA complementarity. We also define tiling recognizable Watson-Crick local languages and prove some closure properties. We also extend Watson-Crick local languages to finite arrays and prove some closure properties.