We consider the statistical inference for noisy incomplete 1-bit matrix M. Despite the importance of uncertainty quantification to matrix completion, most of the categorical matrix completion literature focus on point estimation and prediction. This paper moves one step further towards the statistical inference for 1-bit matrix completion. Under a popular nonlinear factor analysis model, we obtain a point estimator and derive its asymptotic distribution for any linear form of M and latent factor scores. Moreover, our analysis adopts a flexible missing-entry design that does not require a random sampling scheme as required by most of the existing asymptotic results for matrix completion. The proposed estimator is statistically efficient and optimal, in the sense that the Cramer-Rao lower bound is achieved asymptotically for the model parameters. Two applications are considered, including (1) linking two forms of an educational test and (2) linking the roll call voting records from multiple years in the United States senate.