Dynamic Spectrum Access allows wireless users to access a wide range of spectrum which increases a node's ability to communicate with its neighbors, and spectral efficiency through opportunistic access to licensed bands. Our study focuses on the scalability of network performance, which we define in terms of network transport capacity and end-to-end throughput per node, as the network density increases. We develop an analytical procedure for performance evaluation of ad hoc DSA networks using Markov models, and analyze the performance of a DSA network with one transceiver per node and a dedicated control channel. We also develop and integrate a detailed model for energy detection in Poisson networks with sensing. We observe that the network capacity scales sub-linearly with the number of DSA users and the end-to-end throughput diminishes, when the number of data channels is fixed. Nevertheless, we show that DSA can improve network performance by allowing nodes to access more spectrum bands while providing a mechanism for spectrum sharing and maintaining network wide connectivity. We also observe that the percentage of relative overhead at the medium access layer does not scale with the number of users. Lastly, we examine the performance impact of primary user density, detection accuracy, and the number of available data channels. The results help to answer the fundamental question of the scaling behavior of network capacity, end-to-end throughput, and network overhead in ad hoc DSA networks.