Agentic AI networking (AgentNet) is a novel AI-native networking paradigm in which a large number of specialized AI agents collaborate to perform autonomous decision-making, dynamic environmental adaptation, and complex missions. It has the potential to facilitate real-time network management and optimization functions, including self-configuration, self-optimization, and self-adaptation across diverse and complex environments. This paper proposes SANet, a novel semantic-aware AgentNet architecture for wireless networks that can infer the semantic goal of the user and automatically assign agents associated with different layers of the network to fulfill the inferred goal. Motivated by the fact that AgentNet is a decentralized framework in which collaborating agents may generally have different and even conflicting objectives, we formulate the decentralized optimization of SANet as a multi-agent multi-objective problem, and focus on finding the Pareto-optimal solution for agents with distinct and potentially conflicting objectives. We propose three novel metrics for evaluating SANet. Furthermore, we develop a model partition and sharing (MoPS) framework in which large models, e.g., deep learning models, of different agents can be partitioned into shared and agent-specific parts that are jointly constructed and deployed according to agents' local computational resources. Two decentralized optimization algorithms are proposed. We derive theoretical bounds and prove that there exists a three-way tradeoff among optimization, generalization, and conflicting errors. We develop an open-source RAN and core network-based hardware prototype that implements agents to interact with three different layers of the network. Experimental results show that the proposed framework achieved performance gains of up to 14.61% while requiring only 44.37% of FLOPs required by state-of-the-art algorithms.