We consider in this paper a networked system of opinion dynamics in continuous time, where the agents are able to evaluate their self-appraisals in a distributed way. In the model we formulate, the underlying network topology is described by a strongly connected graph. For each ordered pair of adjacent agents (i, j), we assign a function of self-appraisal to agent i, which measures the level of importance of agent i to agent j. Thus by communicating only with his neighbors, each agent is able to calculate the difference between his level of importance to others and others' level of importance to him. The dynamical system of self-appraisals is then designed to drive these differences to zero. We show that for almost all initial conditions, the trajectory of the dynamical system asymptotically converges to an equilibrium which is exponentially stable.