The class of generalized satisfiability problems, first introduced by Schaefer in 1978, presents a uniform way of studying the complexity of Boolean constraint satisfaction problems with respect to the nature of constraints allowed in the input. We investigate the complexity of local search for this class of problems. We prove a dichotomy result: any generalized satisfiability local search problem is either in P or PLS-complete. In the meantime our study contributes to a better understanding of the complexity class PLS through the identification of an appropriate tool that captures reducibility among Boolean constraint satisfaction local search problems: sensitive implementation.