  Although a lot of approaches are developed to release network data with a
differentially privacy guarantee, inference using noisy data in many network
models is still unknown or not properly explored. In this paper, we release the
bi-degree sequences of directed networks using the Laplace mechanism and use
the $p_0$ model for inferring the degree parameters. The $p_0$ model is an
exponential random graph model with the bi-degree sequence as its exclusively
sufficient statistic. We show that the estimator of the parameter without the
denoised process is asymptotically consistent and normally distributed. This is
contrast sharply with some known results that valid inference such as the
existence and consistency of the estimator needs the denoised process. Along
the way, a new phenomenon is revealed in which an additional variance factor
appears in the asymptotic variance of the estimator when the noise becomes
large. Further, we propose an efficient algorithm for finding the closet point
lying in the set of all graphical bi-degree sequences under the global $L_1$
optimization problem. Numerical studies demonstrate our theoretical findings.
