Molecular computing executed via local interactions of spatially contiguous set of molecules has potential advantages of (i) speed due to increased local concentration of reacting species, (ii) sharper switching behavior due to single molecule interactions, (iii) parallelism since each circuit operates independently of another and (iv) modularity and scalability due to the ability to reuse DNA sequences in spatially separated regions. We propose detailed designs for local molecular computations that involve spatially contiguous molecules arranged on addressable substrates. The circuit acts via enzyme-free DNA hybridization reaction cascades. Our designs include composable OR, AND, propagation Boolean gates and techniques to achieve higher degree fan-in and fan-out. A biophysical model of localized hybridization reactions is used to estimate the effect of locality on reaction rates. We also use the Visual DSD simulation software in conjunction with these localized reaction rates to simulate a localized circuit for computing the square root of a four bit number.