This paper investigates a multi-user wireless powered fog computing (FC) network, where energy-limited wireless sensor devices (WSDs) first harvest energy from a nearby hybrid access point (HAP) and then, compute their tasks locally (i.e., the local computing (LC) mode) or offload the tasks to the HAP (i.e., the FC mode) via a binary offloading policy. An optimization problem is formulated to minimize the transmit power at the HAP by jointly optimizing the time allocation ratio and the computing mode selection vector, under the energy causality constraints and the WSDs computing rate requirements constraints. To efficiently solve the formulated non-convex problem in a distributed manner, an alternating direction method of multipliers (ADMM)-based algorithm is designed. With our presented ADMM-based algorithm, each WSD is able to optimize its computing mode and offloading time with local channel state information (CSI). For comparison, a channel-sorting-based (CSB) centralized algorithm with global CSI is also designed. Simulation results show that the proposed distributed algorithm achieves the comparable performance with the centralized algorithm or by using the exhaustive search one. Moreover, to minimize the transmit power, the WSDs with the better channel quality are inclined to select the LC mode, which is much different from traditional design.