This paper presents a distributed cell-free communication and radar system that operates in the uplink. The system schedules dedicated transmit (Tx) access points (APs) to transmit dedicated radar signals in the uplink together with the user equipment (UE). The receiving (Rx) APs decode the UE payloads while also detecting targets based on the Tx AP signals. To mitigate the added Tx AP interference, the Rx APs use multiuser processing to recover the UE payloads, while a combination of large processing gains, adaptive beamforming, spatial diversity, interference cancellation and power control is used to mitigate the UE interference impacting the radar. The radar introduces few changes to the physical layer and the additional computations needed are comparable to the communication system. The system is validated numerically by using Monte-Carlo simulations, where we highlight the inherent trade-offs between the various system parameters (such as the power control balancing, and the number of Tx APs scheduled and UEs cancelled) and show that both the communication and radar systems can be effectively integrated into the same network at a near optimal performance.