The information necessary for a fix is broadcast. The locally stored database helps improve the time to get a fix. A GPS receiver, going from a cold start, needs to listen for many minutes to acquire and decode enough signals to have the required satellite position and timing information to do the calculation.

Most of the power consumption is for the radio reception that has to detect and decode signals from multiple constantly shifting sources, dealing with their very low signal-to-noise ratios and other challenges like multipath distortion due to atmosphere and surface reflections.

It's pretty remarkable how much miniaturization has improved the efficiency of these radios. E.g. going from the early "portable" GPS units that essentially had a lead-acid car or motorcycle battery to today's wearables that run on a tiny power budget while supporting a wider range of satellite constellations and radio bands.