Accurate, robust, and adaptive localization is essential for various robotic operations. This paper proposes a parametric message passing (MP) algorithm for distributed collaborative localization, unifying Gaussian belief propagation (GBP) for robot state estimation and mean-field (MF) approximation for noise inference. Unlike conventional MP algorithms, the proposed formulation directly optimizes the natural parameters of beliefs, establishing a principled connection between MP and gradient estimators for handling non-conjugate terms arising from nonlinear measurement models. Beyond linearization and sampling, we further design a normalizing flow (NF)-based gradient estimator, enabling learnable sampling. End-to-end training tunes NF parameters according to the behavior of MP, improving estimation accuracy while reducing the required number of iterations. To support robot states involving rotations, the method is extended to Lie group spaces. Finally, the method is applied to a multirobot localization task fusing odometry, global navigation satellite system (GNSS) measurements, and inter-robot ultra wideband (UWB) ranging. Extensive simulations and experiments on autonomous surface vehicles (ASVs) demonstrate superior state estimation accuracy, enhanced robustness and adaptability under ranging outliers and varying noise statistics, as well as good generalization across different operating conditions.

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