By virtue of the low cost and flexible deployment, unmanned aerial vehicles (UAVs) are becoming the most promising platform for active radar sensing to find targets in shaded areas and build a temporary communication infrastructure at disaster sites. Radar and wireless communication are essential functions for reliable control of UAVs and real-time exchange of sensed measurements. However, due to the limited payload and power budget of UAVs, it is impossible to mount two individual radar and communication systems, especially on small- or medium-sized UAVs. Recently, joint communication and radar (JCR) systems have emerged as a breakthrough, which perform the communication and radar functions simultaneously on the same hardware platform. Compared to a simple combination of individual communication and radar systems, a UAV JCR system can minimize the implementation cost, power consumption, payload, and radio resource usage. This paper proposes transmit beamformer and receive beamformer designs for a UAV network composed of a UAV JCR base station, a target, and multiple user equipments. Specifically, the UAV transmits an orthogonal frequency division multiplexing (OFDM)-modulated waveform with transmit and receive beamforming to perform multi-input multi-output (MIMO) radar and multiuser-MIMO (MU-MIMO) communication functions simultaneously. Unlike the previous OFDM radar scheme that employs the communication symbol repetition method, a novel receive beamformer design is proposed to resolve the inter-carrier interference problem in the range/velocity estimation process, which therefore does not sacrifice the communication transmission rate. As a result, simulation results show that the proposed scheme achieves a higher radar signal-to-clutter ratio while achieving a higher communication sum-rate than existing schemes.