This paper addresses the problem of simultaneous distributed state estimation and control of linear systems with linear state feedback, subjected to process and measurement noise, under the constraints of quantized and rate-limited network data transmission. In the set-up adopted, sensors and actuators communicate through a network with a strongly connected topology. Unlike the case of centralized linear systems, for which the separation principle holds, the above practical assumption prevents the separate design of observers and controller because each of the nodes does not necessarily have access to the control inputs generated at all the other nodes. We derive a linear distributed Luenberger observer and derive a set of sufficient conditions that guarantee ultimate boundedness of the estimation error and system state vectors, with bounds that depend on the L1 norm of the noise signals and the number of bits used in the transmissions. A numerical example illustrates the performance and effectiveness of the proposed algorithm in controlling a network of open-loop unstable systems.