How is linguistic communication possible? How do we come to share the same meanings of words and utterances? One classical position holds that human beings share a transcendental “platonic” ideality independent of individual cognition and language use (Frege 1948). Another stresses immanent linguistic relations (Saussure 1959), and yet another basic embodied structures as the ground for invariant aspects of meaning (Lakoff and Johnson 1999). Here we propose an alternative account in which the possibility for sharing meaning is motivated by four sources of structural stability: 1) the physical constraints and affordances of our surrounding material environment, 2) biological constraints of our human bodies, 3) social normative constraints of culture and society, and 4) the local history of social interactions. These structures and constraints interact in dynamical ways in actual language usage situations: local dialogical and social dynamics motivate and stabilize the profiling of a conceptual space already highly structured by our shared biology, culture, and environment. We will substantiate this perspective with reference to recent studies in experimental pragmatics and semiotics in which participants interact linguistically to solve cooperative tasks. Three main cases will be considered: The dynamic grounding of linguistic categories, the construction of conceptual models to relate entities in a scene, and the construction of shared conceptual scales for assessing and appraising subjective experiences.