Device-to-device (D2D) discovery is the inextricable prelude for the direct exchange of local traffic between cellular users in proximity. The D2D discovery process can be based on either autonomous actions taken by D2D-enabled devices, also known as network-assisted D2D discovery or core network functionalities to estimate proximity, also known as network-assisted D2D discovery. A key advantage of network-assisted D2D discovery is its potential to reduce the energy, signaling, and interference required for D2D discovery, by exploiting knowledge of the network layout. We analyze the performance of network-assisted D2D discovery in random spatial networks and derive useful guidelines for its design. We derive approximate expressions for the distance distribution between two D2D peers conditioned on the core network's knowledge of the cellular network layout, assuming that the base stations are distributed according to the Poisson point process. The expressions are used to assess the interplay between the D2D discovery probability and key system parameters, such as network intensity and transmit power, as well as to identify conditions to maximize the D2D discovery probability. Numerical results validate the accuracy of our findings and provide insights on the performance tradeoffs of network-assisted D2D discovery.