Language models such as GPT and Llama have shown remarkable ability on diverse natural language tasks, yet their performance on complex table tasks (e.g., NL-to-Code and data cleaning) remains suboptimal. Improving performance typically requires task-specific fine-tuning, which depends on expensive human labeling and is prone to overfitting. In this work, we propose Table-LLM-Specialist, a self-trained fine-tuning paradigm designed for table tasks. Our key insight is that many table tasks admit two dual formulations: a generative version and a classification version. Leveraging this duality, we introduce a Generator-Validator paradigm that iteratively generates and validates training data using language models, enabling effective fine-tuning without manually labeled data. Extensive evaluations on Llama, GPT-3.5, and GPT-4 show that Table-LLM-Specialist achieves (1) strong performance across diverse tasks compared to base models, for example, models fine-tuned on GPT-3.5 often surpass GPT-4 level quality; (2) lower deployment cost by enabling smaller models to reach high quality with reduced latency and cost; and (3) better generalization across multiple benchmarks, due to training on diverse, systematically generated data from real-world tables. Our code is available at https://github.com/microsoft/Table-Specialist. Models fine-tuned with Table-LLM-Specialist have been integrated into Microsoft Excel and are deployed in production for automated table data cleaning.