Molecular communication (MC) is a promising paradigm for conveying information in nanonetworks, where small particles like molecules are the carriers of information. In this paper, we investigate a special communication scenario, where the MC system consists of a pair of mobile transmitter and receiver nanomachines. Both of them move randomly in a free diffusion manner. Due to the randomness generated by the motion of nanomachines and particle counting noise, the received signal at the receiver nanomachine, i.e., the number of molecules within the volume of the receiver, is a stochastic process on observing time. The statistical analysis of the received signal is provided. The closed-form expressions of the mean and variance of the received signal are derived by considering two kinds of randomness mentioned above. Besides, the distribution of the received signal is investigated. The probability mass function (PMF) of the received signal is given by generalized Gauss-Laguerre quadrature, a numerical analysis method. Furthermore, to obtain the closed-form expression of the PMF of the received signal, we prove that under certain conditions, the received signal follows a Poisson-lognormal distribution and can be approximated by a Poisson distribution. Another interest in this paper is the error performance of the mobile MC systems. Considering the effect of inter-symbol interference (ISI) and noise on detection process, an analytical expression for the probability of error of a simple detector is derived by Gaussian approximation. The accuracy of the proposed analytical expressions and distributions is validated via particle-based simulations.