Although the theoretical properties in the $p_0$ model based on a differentially private bi-degree sequence have been derived, it is still lack of a unified theory for a general class of directed network models with the $p_{0}$ model as a special case. We use the popular Laplace data releasing method to output the bi-degree sequence of directed networks, which satisfies the private standard--differential privacy. The method of moment is used to estimate unknown parameters. We prove that the differentially private estimator is uniformly consistent and asymptotically normal under some conditions. Our results are illustrated by the Probit model. We carry out simulation studies to illustrate theoretical results and provide a real data analysis.