Distributed state estimation in smart grid highly relies on the availability of measurements. Transmitting a lot of measurements within a small time interval is costly and sometimes even impossible. This paper explores the problem of distributed state estimation in smart grid with constraint on the number of measurements that is able to be transmitted in one step. It is shown that there exists a lower bound which depends on the structure of the grid such that if the number of permissible measurements is beyond the bound, then the estimator achieves the same performance as its peer without the constraint. Further, if the number of permissible measurements is below the lower bound, a tradeoff between the performance of the estimator and the measurements transmitted is needed to meet the constraint. A method to attain the tradeoff is offered in this paper. The proposed conclusions and methods are illustrated in the simulation on the IEEE 14-bus system.