Recent research and development interests deal with metasurfaces for wireless systems beyond their consideration as intelligent tunable reflectors. Among the latest proposals is the simultaneously transmitting (a.k.a. refracting) and reflecting reconfigurable intelligent surface (STAR-RIS) which intends to enable bidirectional indoor-to-outdoor, and vice versa, communications thanks to its additional refraction capability. This double functionality provides increased flexibility in concurrently satisfying the quality-of-service requirements of users located at both sides of the metasurfaces, for example, the achievable data rate and localization accuracy. In this paper, we focus on STAR-RIS-empowered simultaneous indoor and outdoor three-dimensional (3D) localization, and study the fundamental performance limits via Fisher information analyses and Cramér Rao lower bounds (CRLBs). We also devise an efficient localization algorithm based on an off-grid compressive sensing (CS) technique relying on atomic norm minimization (ANM). The impact of the training overhead, the power splitting at the STAR-RIS, the power allocation between the users, the STAR-RIS size, the imperfections of the STAR-RIS-to-BS channel, as well as the role of the multi-path components on the positioning performance are assessed via extensive computer simulations. It is theoretically showcased that high-accuracy, up to centimeter level, 3D localization can be simultaneously achieved for indoor and outdoor users, which is also accomplished via the proposed ANM-based estimation algorithm.