The $p_0$ model is an exponential random graph model for directed networks with the bi-degree sequence as the exclusively sufficient statistic. It captures the network feature of degree heterogeneity. The consistency and asymptotic normality of a differentially private estimator of the parameter in the private $p_0$ model has been established. However, the $p_0$ model only focuses on binary edges. In many realistic networks, edges could be weighted, taking a set of finite discrete values. In this paper, we further show that the moment estimators of the parameters based on the differentially private bi-degree sequence in the weighted $p_0$ model are consistent and asymptotically normal. Numerical studies demonstrate our theoretical findings.