To get good precision you need to remove the effect of electrons in the ionosphere. These basically change the index of refraction and therefore the propagation speed of the radio waves. The delayed signal makes your receiver think it is further away from the satellite than it actually is. One way to estimate this delay (instead of relying e.g. on the correction signal of a differential-GPS base station close to you) is to receive signals from the satellites at two different frequencies. The time delay is proportional to f^-2. Knowing the frequencies and the delay between the two signals allows you to estimate the electron content and and correct both signal to the undelayed "infinite frequency" arrival time.

AFAIK each satellite runs two atomic clocks one being much more accurate. Completely different from GPS and way more advanced.

Unsure how it works singal-wise. I assume not only you need different receiver, but also its hardware has to have much higher clock rate.