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During the past two decades, multi-agent optimization problems have drawn increased attention from the research community. When multiple objective functions are present among agents, many works optimize the sum of these objective functions. However, this formulation implies a decision regarding the relative importance of each objective function. In fact, optimizing the sum is a special case of a multi-objective problem in which all objectives are prioritized equally. In this paper, a distributed optimization algorithm that explores Pareto optimal solutions for non-homogeneously weighted sums of objective functions is proposed. This exploration is performed through a new rule based on agents' priorities that generates edge weights in agents' communication graph. These weights determine how agents update their decision variables with information received from other agents in the network. Agents initially disagree on the priorities of the objective functions though they are driven to agree upon them as they optimize. As a result, agents still reach a common solution. The network-level weight matrix is (non-doubly) stochastic, which contrasts with many works on the subject in which it is doubly-stochastic. New theoretical analyses are therefore developed to ensure convergence of the proposed algorithm. This paper provides a gradient-based optimization algorithm, proof of convergence to solutions, and convergence rates of the proposed algorithm. It is shown that agents' initial priorities influence the convergence rate of the proposed algorithm and that these initial choices affect its long-run behavior. Numerical results performed with different numbers of agents illustrate the performance and efficiency of the proposed algorithm.