Summary: A standing challenge in data privacy is the trade-off between the level of privacy and the efficiency of statistical inference. Here, we conduct an in-depth study of this trade-off for parameter estimation in the \(\beta\)-model [S. Chatterjee et al., Ann. Appl. Probab. 21, No. 4, 1400–1435 (2011; Zbl 1234.05206)] for edge differentially private network data released via jittering [V. Karwa et al., J. R. Stat. Soc., Ser. C, Appl. Stat. 66, No. 3, 481–500 (2017; doi:10.1111/rssc.12185)]. Unlike most previous approaches based on maximum likelihood estimation for this network model, we proceed via the method of moments. This choice facilitates our exploration of a substantially broader range of privacy levels – corresponding to stricter privacy – than has been to date. Over this new range, we discover our proposed estimator for the parameters exhibits an interesting phase transition, with both its convergence rate and asymptotic variance following one of three different regimes of behavior depending on the level of privacy. Because identification of the operable regime is difficult, if not impossible in practice, we devise a novel adaptive bootstrap procedure to construct uniform inference across different phases. In fact, leveraging this bootstrap we are able to provide for simultaneous inference of all parameters in the \(\beta\)-model (i.e., equal to the number of nodes), which, to our best knowledge, is the first result of its kind. Numerical experiments confirm the competitive and reliable finite sample performance of the proposed inference methods, next to a comparable maximum likelihood method, as well as significant advantages in terms of computational speed and memory.