That would be a monumental scalability problem. Given the rapid expansion of GPS usage, you'd basically have to launch Google into orbit, if the system required client-server interaction with the satellites.
GPS is way cooler than that. GPS is receive-only. It listens to the satellites as they transmit exact times and positions, and it calculates its position by correlating the readings from a minimum of four satellites. Four equations in four unknowns: latitude, longitude, elevation, and time. Actually, it could do with only three satellites, but the fourth is necessary if the GPS receiver lacks its own synchronized, miniaturized, dirt-cheap atomic clock (LOL) — it needs to work out its own time to the nanosecond in order for the concept to work. Additional satellites beyond the minimum of four add to accuracy.
A fourth constraint can be supplied if you know your elevation above the Geoid. (Each equation supplying satellite location and time delay is a constraint.) I don’t know if you are familiar with Bankcroft’s method, but it is easily adaptable to knowledge of your distance from the center of the Geiod (earth), or have exact knowledge of time GPS time.
BTY, the more satellites (or other constraints, e.g., altimeter data) you have, the more accurate you estimate of position.