Both, the network configuration and the way of terrestrial reference frame (TRF) realization, affect the global geodetic products delivered from the Global Navigation Satellite Systems (GNSS) data processing. The purpose of this study is to analyze the differences in GNSS products, such as station coordinates, Earth rotation parameters, geocenter coordinates (GCC), and satellite orbits delivered from the double-difference multi-GNSS (GPS, GLONASS, and Galileo) processing, which may arise from (1) using a homogeneous and inhomogeneous network of multi-GNSS stations, (2) different approaches to the TRF realization using minimum constraint conditions, and (3) different approaches to handling of GCC in GNSS global processing. The questionable quality of GCC delivered from the global GNSS solutions is described with a special attention to network effects and system-specific parameters. We found that Galileo can provide GCC, whose quality corresponds to the GPS series. Moreover, the GCC from Galileo is of a better quality than those based on GLONASS data, despite the same number of nominal orbital planes and a much lower number of active satellites. When the No-Net-Translation constraint is not applied on the GNSS network, the station coordinate repeatability is worsened by about 70%, 55%, and 25% for the north, east, and up components, respectively, compared to the solution when applying No-Net-Translation and when having the network origin consistent with the international TRF. We thus infer that the No-Net-Translation condition is mandatory in global GNSS solutions.