Distributed Constraints Optimization (DCOP) is a powerful framework for representing and solving distributed combinatorial problems, where the variables of the problem are owned by different agents. DCOP algorithms search for the optimal solution, optimizing the total gain (or cost) that is composed of all gains of all agents. Local search (LS) DCOP algorithms search locally for an approximate such solution. Many multi-agent problems include constraints that produce different gains (or costs) for the participating agents. Asymmetric gains of constrained agents cannot be naturally represented by the standard DCOP model. The present paper proposes a general framework for Asymmetric DCOPs (ADCOPs). The new framework is described and its differences from former attempts are discussed. New local search algorithms for ADCOPs are introduced and their advantages over existing algorithms and over former representations are discussed in detail. The new proposed algorithms for the ADCOP framework are evaluated experimentally and their performance compared to existing algorithms. Two measures of performance are used: quality of solutions and loss of privacy. The results show that the new algorithms significantly outperform existing DCOP algorithms with respect to both measures.