In overlapping spectrum sharing, due to the complexity of cognitive environment, it is a real challenge for a secondary user (SU) to correctly sense the usage of the spectrum in real time. To tackle this challenge, a social awareness-aided transmit power control policy for SUs is developed. First, a social network composed of a group of third-party sensing nodes that do not share the spectrum with the PU is established, which helps an SU collect the power information of the PU. Then, we design a Dueling Deep Q-Network (DQN) model to achieve efficient dynamic spectrum sharing between the PU and the SU with the power information collected in the social network. Experimental results show that the spectrum sharing success rate is higher and the comprehensive performance is improved with the sensing nodes selected by the social relationship. Moreover, compared with other deep reinforcement learning (DRL) algorithms, the performance of Dueling DQN is more stable on our targeted spectrum sharing problem.