There are rapidly emerging needs to deal with distributed sources of data (sensors and sensor networks, web sites, databases). While recognizing their limited accessibility at a global level (associated with technical constraints and/or privacy issues) and fully acknowledging benefits of collaborative processing, we propose a concept of Collaborative Computational Intelligence (CI), and collaborative fuzzy models, in particular. The variety of possible mechanisms of interaction is organized into a setting of the C3 interaction paradigm (communication - collaboration - consensus). This helps us offer a coherent taxonomy of various schemes of interaction which in the sequel implies a certain development of a suite of algorithms. In this setting, the role granular information in the establishing of the mechanisms of interaction plays a pivotal role. We consider distributed fuzzy models and fuzzy modeling. In particular, we elaborate on the key design issues concerning fuzzy rule-based systems with local functional models occurring at their conclusion parts and show how the fundamental modes of interaction are exploited here. It will be demonstrated that more advanced constructs such as type-2 fuzzy sets emerge naturally in distributed fuzzy modeling and come with a well-defined semantics of their membership functions by being fully reflective of the character of the underlying distributed data. In the context of collaborative fuzzy modeling, we bring forward a concept experience-consistent fuzzy system identification showing how fuzzy models built on a basis of limited data can benefit from taking advantage of the past experience conveyed in the form of previously constructed fuzzy models. Detailed algorithmic considerations embrace several design scenarios in which we apply the mechanism of experience consistency at the level of conditions and conclusions of the rules. We also show that a level of achieved experience-driven consistency can be quantified through fuzzy sets (fuzzy numbers) of the parameters of the local models standing in the conclusion parts of the rules this leading to the emergence of granular constructs of fuzzy modeling.