In many emerging applications, multiple sensors transmit their measurements to a remote estimator over a shared medium. In such a system, the optimal sampling rates at each sensor depend on the nature of the stochastic process being observed as well as the available communication capacity. Our main contribution is to show that the problem of determining optimal sampling rates may be posed as a network utility maximization problem and solved using appropriate modifications of the standard dual decomposition algorithms for network utility maximization. We present two such algorithms, one synchronous and one asynchronous, and show that under mild technical conditions, both algorithms converge to the optimal rate allocation. We present a detailed simulation study to illustrate that the asynchronous algorithm is able to adapt the sampling rate to change in the number of sensors and the available channel capacity and is robust to packet drops.