In wireless peer-to-peer networks that serve various proximity-based applications, peer discovery is the key to identifying other peers with which a peer can communicate and an understanding of its performance is fundamental to the design of an efficient discovery operation. This paper analyzes the performance of wireless peer discovery through comprehensively considering the wireless channel, spatial distribution of peers, and discovery operation parameters. The average numbers of successfully discovered peers are expressed in closed forms for two widely used channel models, i.e., the interference limited Nakagami-m fading model and the Rayleigh fading model with nonzero noise, when peers are spatially distributed according to a homogeneous Poisson point process. These insightful expressions lead to the design principles for the key operation parameters including the transmission probability, required amount of wireless resources, level of modulation and coding scheme (MCS), and transmit power. Furthermore, the impact of shadowing on the spatial performance and suggested design principles is evaluated using mathematical analysis and simulations.