We consider the problem of enabling a network of agents to estimate the state of a discrete-time nonlinear dynamical system. At each time step, each agent in the network receives a measurement characterized by a nonlinear function of the system state and exchanges information with its neighbors in the network. We propose an optimization-based estimator where agents collaboratively solve a distributed optimization problem while satisfying a communication constraint in the form of a fixed number of distributed optimization iterations at each estimation time step. Subject to the assumptions that the system is collectively observable, and the communication network is time-varying and strongly connected, we show that for any given $\lambda$ which satisfies $0 < \lambda < 1$, it is possible to choose $q$, the number of the distributed optimization iterations, so that the estimation error for each agent converges to zero at least as fast as $\lambda^{t}$ does.