Cognitive radio (CR) is a key enabler realizing future networks to achieve higher spectral efficiency by allowing spectrum sharing between different wireless networks. It is important to explore whether spectrum access opportunities are available, but conventional CR based on transmitter (TX) sensing cannot be used to this end because the paired receiver (RX) may experience different levels of interference, according to the extent of their separation, blockages, and beam directions. To address this problem, this paper proposes a novel form of medium access control (MAC) termed sense-and-predict (SaP), whereby each secondary TX predicts the interference level at the RX based on the sensed interference at the TX; this can be quantified in terms of a spatial interference correlation between the two locations. Using stochastic geometry, the spatial interference correlation can be expressed in the form of a conditional coverage probability, such that the signal-to-interference ratio (SIR) at the RX is no less than a predetermined threshold given the sensed interference at the TX, defined as an opportunistic probability (OP). The secondary TX randomly accesses the spectrum depending on OP. We optimize the SaP framework to maximize the area spectral efficiencies (ASEs) of secondary networks while guaranteeing the service quality of the primary networks. Testbed experiments using USRP and MATLAB simulations show that SaP affords higher ASEs compared with CR without prediction.