Typical Global Navigation Satellite System (GNSS) receivers offer precision in the order of meters. This error margin is excessive for vehicular safety applications, such as forward collision warning, autonomous intersection management, or hard braking sensing. In this work we develop a Cooperative GNSS Positioning System (CooPS) that uses Vehicle to Vehicle (V2V) and Vehicle to Infrastructure (V2I) communications to cooperatively determine absolute and relative position of the ego-vehicle with enough precision. To that end, we use differential GNSS through position vector differencing to acquire track and across-track axes projections, employing elliptical and spherical geometries. We evaluate CooPS performance by carrying out real experiments using off-the-shelf IEEE 802.11p equipment at the campus of the Federal University of Rio de Janeiro. We obtain an accuracy level under 1.0 and 1.5 m for track (where-in-lane) and across-track (which-lane) axes, respectively. These accuracy levels were achieved using a 2.5 m accuracy circular error probable (CEP) of 50% and a 5 Hz navigation update rate GNSS receiver.