Plastic metals and low-dimensional materials are extensively utilized as reinforcements in fabricating bio-inspired staggered composites. Here, we introduce a comprehensive analytical model to investigate the influence of reinforcement plasticity on the mechanical properties of staggered composites while preserving the non-linear plastic characteristics of the matrix. Competitive plastic deformation in both the reinforcement and the matrix leads to two distinct deformation modes: reinforcement-first yield or matrix-first yield. Each mode exhibits different stages of deformation and failure in plastic staggered composites. Our analytical formulae, validated via finite element analysis, establish connections between effective stress and strain responses, material compositions, and structural geometry, thereby revealing non-linear shear stress transfer and plastic evolution mechanisms. Furthermore, we discover that tailoring the plasticity of the reinforcement while maintaining the dominant plastic deformation of the matrix, can overcome the trade-off between composite strength and ductility. Our model provides valuable insights into designing high-performance metal-reinforced staggered composites and can be further extended to explore the mechanical properties of plastic low-dimensional material-reinforced nanocomposites with noncovalent interfaces.