The evolution of 6G networking toward agentic AI networking (AgentNet) systems requires a shift from traditional data pipelines to task-aware, agentic AI-native communication solutions. Emergent communication, a novel communication paradigm in which autonomous agents learn their own signaling protocols through interaction, is increasingly viewed as a promising solution to address the challenges posed by existing rigid, predefined protocol-based networking architecture. However, most existing emergent communication frameworks fail to account for physical networking constraints, such as bandwidth and computational complexity, and often lack a rigorous information-theoretical foundation. To address these challenges, this paper introduces a novel emergent communication framework that facilitates collaborative task-solving among heterogeneous agents through an information-theoretic lens. We propose a novel joint loss function that unifies the optimization of decision-making functions and the learning of communication signaling. Our proposed solution is grounded on the multi-agent and multi-task distributed information bottleneck (DIB) theory, which allows the quantification of the fundamental trade-off between task-relevant information representation and computational complexity. We further provide theoretical generalization bounds of the emergent communication protocol during decentralized inference across unseen environmental states. Experimental validation on a real-world hardware prototype confirms that our proposed framework significantly improves generalization performance, compared to the state-of-the-art solutions.