This paper studies the opportunistic spectrum access (OSA) of the secondary users in a large-scale overlay cognitive radio (CR) network. Two threshold-based OSA schemes, namely the primary receiver assisted (PRA) protocol and the primary transmitter assisted (PTA) protocol, are investigated. Under the PRA/PTA protocol, a secondary transmitter (ST) is allowed to access the spectrum only when the maximum signal power of the received beacons/pilots sent from the active primary receivers/transmitters (PRs/PTs) is lower than a certain threshold. To measure the resulting transmission opportunity for the secondary users by the proposed OSA protocols, the concept of spatial opportunity, which is defined as the probability that an arbitrary location in the primary network is detected as a spatial spectrum hole, is introduced and then evaluated by applying tools from stochastic geometry. Based on spatial opportunity, the coverage (non-outage transmission) performance in the overlay CR network is analyzed. With the obtained results of spatial opportunity and coverage probability, we finally characterize the spatial throughput, which is defined as the average spatial density of successful transmissions in the primary/secondary network, under the PRA and PTA protocols, respectively.