The emergence of Agentic AI is fundamentally transforming how software is designed, developed, and maintained. Traditional software development methodologies such as Agile, Kanban, ShapeUp, etc, were originally designed for human-centric teams and are increasingly inadequate in environments where autonomous AI agents contribute to planning, coding, testing, and continuous learning. To address this methodological gap, we present "Agentsway" a novel software development framework designed for ecosystems where AI agents operate as first-class collaborators. Agentsway introduces a structured lifecycle centered on human orchestration, and privacy-preserving collaboration among specialized AI agents. The framework defines distinct roles for planning, prompting, coding, testing, and fine-tuning agents, each contributing to iterative improvement and adaptive learning throughout the development process. By integrating fine-tuned LLMs that leverage outputs and feedback from different agents throughout the development cycle as part of a retrospective learning process, Agentsway enhances domain-specific reasoning, and explainable decision-making across the entire software development lifecycle. Responsible AI principles are further embedded across the agents through the coordinated use of multiple fine-tuned LLMs and advanced reasoning models, ensuring balanced, transparent, and accountable decision-making. This work advances software engineering by formalizing agent-centric collaboration, integrating privacy-by-design principles, and defining measurable metrics for productivity and trust. Agentsway represents a foundational step toward the next generation of AI-native, self-improving software development methodologies. To the best of our knowledge, this is the first research effort to introduce a dedicated methodology explicitly designed for AI agent-based software engineering teams.