Reliable precipitation monitoring is essential for disaster risk reduction, water resources management, and agricultural decision-making. Multi-source satellite observations, particularly the combination of geostationary infrared and passive microwave measurements, have become a primary means of precipitation detection. Traditional multi-source satellite precipitation estimation methods remain computationally inefficient, and many deep learning methods lack the flexibility to incorporate new sensors without retraining the full model. Here we introduce PRISMA (Precipitation Inference from Satellite Modalities via generAtive modeling), a plug-and-play latent generative framework for multi-sensor precipitation estimation. PRISMA learns an unconditional precipitation prior from IMERG Final fields and constrains it through independently trained, sensor-specific conditional branches, allowing new observation sources to be incorporated without retraining the generative backbone. Applied to FY-4B AGRI infrared and GPM GMI microwave observations, PRISMA improves Critical Success Index by up to 40.3% and reduces root-mean-square error by 22.6% relative to infrared-only estimation within microwave swaths, while also improving probabilistic skill and maintaining an average inference time of about 37 s. Independent rain-gauge validation across China confirms consistent gains, and typhoon case studies show that microwave conditioning restores eyewall and spiral rainband structures, reducing storm-core mean absolute error by up to 42.3%. PRISMA thus provides an extensible and efficient framework for multi-sensor precipitation estimation.