Privacy has traditionally been a major motivation of distributed problem solving. In this paper, we focus on privacy issues when solving Distributed Constraint Optimization Problems (DCOPs) using a local search approach. Two such popular algorithms exist to find good solutions to DCOP: DSA and GDBA. However, these were not designed with privacy in mind. In this paper, we propose DSAB, a new algorithm that merges ideas from both algorithms to allow extensive handling of constraint privacy. We also study how algorithms behave when solving Utilitarian DCOPs, where utilitarian agents want to reach an agreement while reducing the privacy loss. We show experimentally that this allows us reductions of domain privacy loss by a factor 2 to 3 with no significant impact on the quality of the solution.