The advances in low-latency communications networks combined with the paradigm of Intelligent Transportation Systems (ITS) have opened a number of opportunities to develop network-based collaborative positioning and navigation. Recent research works indeed, have fostered the concept of networked Global Navigation Satellite System (GNSS) receivers supporting the sharing of raw measurements with other receivers connected to the network. Such measurements (i.e., pseudorange and Doppler) can be processed through Differential GNSS techniques to retrieve inter-receiver distances which can be in turn integrated to improve positioning performance. This paper investigates an improved Bayesian estimation for a sensorless (i.e. unavailable Inertial Navigation System (INS)), tight-integration of Differential-GNSS-based collaborative measurements through a modified Particle Filter (PF). Differently from Hybrid Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) indeed, a PF natively support the non-Gaussian noise distribution characterizing GNSS-based inter-receiver distances. The proposed PF was hence designed, implemented and optimized according to the architecture of a proprietary INS-free Global Navigation Satellite System (GNSS) software receiver and tested with realistic Radio-Frequency (RF) signals, thus showing remarkable improvement in positioning accuracy.