A wireless power system may include power transmitting devices, power receiving devices, and power transmitting and receiving devices. Control circuitry in the power transmitting and receiving device may determine whether to operate in a power receiving mode or a power transmitting mode. The preferred mode for the power transmitting and receiving device may depend on the types of devices within the system, the state of charge of the batteries of the devices in the system, whether or not one or more devices in the system are connected to a wired power source, etc. Each power transmitting and receiving device may have a default mode. Upon detection of an adjacent device, the wireless power transmitting and receiving device enters a configuration phase in its default mode. Before or during a power transfer phase, the power transmitting and receiving device may swap roles from its default mode to another mode.