We prove the local asymptotic mixed normality (LAMN) property for a family of probability measures defined by parametrized diffusion processes with nonsynchronous observations. We assume that observation times of processes are independent of processes and we will study asymptotics when the maximum length of observation intervals goes to zero in probability. We also prove that the quasi-maximum likelihood estimator and the Bayes-type estimator proposed in Ogihara and Yoshida (Stochastic Process. Appl. 124 (2014) 2954-3008) are asymptotically efficient.