A simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) assisted wireless powered communication network (WPCN) is proposed, where two energy-limited devices first harvest energy from a hybrid access point (HAP) and then use that energy to transmit information back. To fully eliminate the doubly-near-far effect in WPCNs, two STAR-RIS operating protocol-driven transmission strategies, namely energy splitting non-orthogonal multiple access (ES- NOMA) and time switching time division multiple access (TS- TDMA) are proposed. For each strategy, the corresponding optimization problem is formulated to maximize the minimum throughput by jointly optimizing time allocation, user transmit power, active HAP beamforming, and passive STAR-RIS beamforming. For ES-NOMA, the resulting intractable problem is solved via a two-layer algorithm, which exploits the one-dimensional search and block coordinate descent methods in an iterative manner. For TS-TDMA, the optimal active beamforming and passive beamforming are first determined according to the maximum-ratio transmission beamformer. Then, the optimal solution of the time allocation variables is obtained by solving a standard convex problem. Numerical results show that: 1) the STAR-RIS can achieve considerable performance improvements for both strategies compared to the conventional RIS; 2) TS- TDMA is preferred for single-antenna scenarios, whereas ES- NOMA is better suited for multi-antenna scenarios; and 3) the superiority of ES-NOMA over TS-TDMA is enhanced as the number of STAR-RIS elements increases.