Microwave signals can be generated by photodetecting the repetition frequencies of the soliton microcombs. In comparison to other methods, the dual-pumped method allows for the stable generation of the soliton microcombs even with resonators having lower Q-factors. However, introducing an additional pump laser may affect the phase noise of the generated microwave signals when using these dual-pumped soliton microcombs. Here, we investigate the factors that could influence the phase noise of microwave signals generated with dual-pumped soliton microcombs, including the polarization, amplitude noise, and phase noise of the two pumps. We demonstrate a 25.25 (12.63) GHz microwave with phase noise reaching -112(-118) dBc/Hz at a 10 kHz offset frequency, surpassing the performance of previous reports on microwave generation using free-running Si3N4 soliton microcombs, even those generated with higher Q microresonators. We analyze the noise floor of the generated microwave signals and establish a phase noise simulation model to study the limiting factors in our system. Our work highlights the potential of generating low-phase-noise microwave signals using free-running dual-pumped soliton microcombs.