Cooperation between road vehicles through information exchange is a promising way to enhance their absolute and relative positions. This paper presents an approach for generating, sharing and applying Global Navigation Satellite System (GNSS) pseudorange corrections through a V2X communication network. Conventionally, differential corrections are generated by fixed base stations with known positions and sent to mobile users. Here, the proposed cooperative method has no central server and the estimation of the raw measurements errors is done in a fully distributed way. Using a model of the correlation of the pseudorange errors and through the knowledge of the local motions of the vehicles obtained by Dead Reckoning (DR) or tracking, a non linear observability shows that the estimation problem is solvable. A cooperative and fully distributed estimation method is then presented using Set Inversion and Constraint propagation techniques. Positions, pseudorange estimated errors and DR data are shared in the network of vehicles and confidence is handled by intervals, in a bounded error context. This allows computing highly reliable confidence domains with no direct range measurements, which is crucial for applications involving close proximity navigation. Indeed, the proposed data fusion framework does not require any linearization of the equations and is insensitive to the data incest problem since the same information can be exploited several times in the computation process without making the estimation over-converge. Results using real measurements are presented to illustrate the performance of the proposed cooperative method in comparison with standalone estimation. A classical sequential Bayesian method has also been implemented on the same data set and compared in terms of accuracy and confidence with a ground truth system.