Due to the features of on-demand deployment and flexible observation, unmanned aerial vehicles (UAVs) are promising for serving as the next-generation aerial sensors by using their onboard sensing devices. Compared to a single UAV with limited sensing coverage and communication capability, multi-UAV cooperation is able to realize more effective sensing and transmission (S&T) services, and delivers the sensory data to the control center more efficiently for further analysis. Nevertheless, most existing works on multi-UAV sensing mainly focus on mutually exclusive task allocation and independent data transmission, which did not fully exploit the benefit of multi-UAV sensing and communication. Motivated by this, we propose a novel multi-UAV cooperative S&T scheme with overlapped sensing task allocation. Although overlapped task allocation may sound counter-intuitive, it can actually foster cooperative transmission among multiple UAVs through a virtual multi-antenna system and thus reduce the overall sensing mission completion time. To obtain the optimal task allocation and transmit power of the proposed scheme, a mission completion time minimization problem is formulated. To solve this problem, a condition that specifies whether it is necessary for the UAVs to perform overlapped sensing is derived. For the cases of overlapped sensing, this time minimization problem is transformed into a monotonic optimization and is solved by the generic Polyblock algorithm. To efficiently evaluate the mission completion time in each iteration of the Polyblock algorithm, new auxiliary variables are introduced to decouple the otherwise sophisticated joint optimization of transmission time and power. While for the degenerated case of non-overlapped sensing, the closed-form expression of the optimal transmission time is derived, which provides insights into the optimal solution and facilitates the design of an efficient double-loop binary search algorithm to optimally solve the degenerated problem. Finally, simulation results demonstrate that the proposed scheme significantly reduces the mission completion time over benchmark schemes.