Existing API-based agentic systems for RTL code generation are fundamentally misaligned with industrial practice: they assume a golden testbench is available at generation time, rely on closed-source APIs incompatible with chip vendors' air-gapped security requirements, and cannot be trained on vendors' proprietary RTL codebases, leaving valuable internal data unused. Recent self-trained models address the deployment constraint but remain single-turn generators that overlook the critical role of verification in real industrial flows. To bridge these gaps, we present ChipMATE, the first self-trained multi-agent framework for RTL generation. Inspired by industrial practice where correctness emerges from cross-comparison between independently written RTL modules and reference models, ChipMATE pairs a Verilog agent with a Python reference-model agent that mutually verify each other's outputs without any golden oracle. We design a backtrack-based inference workflow to prevent error propagation across turns, and a two-stage training pipeline that first trains each agent individually to saturate its code-generation capability, then trains the team jointly to collaborate effectively. To support the training, we further build a hybrid data-generation framework that produces 64.4K high-quality reference model training samples. ChipMATE achieves 75.0\% and 80.1\% pass@1 on VerilogEval V2 with 4B and 9B base models, outperforming all existing self-trained models and even DeepSeek V4 with 1600B parameters. Our code and model weights are publicly available in https://github.com/zhongkaiyu/ChipMATE.