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Is it possible to abstract a formal mechanism originating schisms and governing the size evolution of social conversations? In this work we propose a constructive solution to this problem: an abstract model of a generic N-party turn-taking conversation. The model develops from simple yet realistic assumptions derived from experimental evidence, abstracts from conversation content and semantics while including topological information, and is driven by stochastic dynamics. We find that a single mechanism, namely the dynamics of conversational party's individual fitness as related to conversation size, controls the development of the self-organized schisming phenomenon. Potential generalizations of the model - including individual traits and preferences, memory effects and more elaborated conversational topologies - may find important applications also in other fields of research, where dynamically-interacting and networked agents play a fundamental role.