Shared autonomy in driving requires anticipating human behavior, flagging risk before it becomes unavoidable, and transferring control safely and smoothly. We propose Diffusion-SAFE, a closed-loop framework built on two diffusion models: an evaluator that predicts multimodal human-intent action sequences for probabilistic risk detection, and a safety-guided copilot that steers its denoising process toward safe regions using the gradient of a map-based safety certificate. When risk is detected, control is transferred through partial diffusion: the human plan is forward-noised to an intermediate level and denoised by the safety-guided copilot. The forward-diffusion ratio $ρ$ acts as a continuous takeover knob-small $ρ$ keeps the output close to human intent, while increasing $ρ$ shifts authority toward the copilot, avoiding the mixed-unsafe pitfall of action-level blending. Unlike methods relying on hand-crafted score functions, our diffusion formulation supports both safety evaluation and plan generation directly from demonstrations. We evaluate Diffusion-SAFE in simulation and on a real ROS-based race car, achieving 93.0%/87.0% (sim/real) handover success rates with smooth transitions.