Both: they broadcast not the location but the orbital characteristics (ephemeris), and devices can save the last received value. As the satellites get 'perturbed' in orbit, their orbital data is updated and re-broadcast.

* https://gssc.esa.int/navipedia/index.php/GPS_Navigation_Mess...

* https://www.e-education.psu.edu/geog862/node/1737

* https://gssc.esa.int/navipedia/index.php/GPS_and_Galileo_Sat...

* https://en.wikipedia.org/wiki/Ephemeris

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.

Yes, satellites broadcast their position and time continuously. There's also the database approach (check A-GPS) where you store the satellite's position and query that but just know that it needs to be updated after a while.

Now about the battery draining - the more satellites your phone GPS captures the higher the precision. You need at least 4 satellites to trilaterate aka get precise lat, long. Listening to the signal from the GPS and then trilaterating is an expensive operation- why? because the satellite signal is very very weak and your phone has to run quite a lot of operations (how far the satellites are, then direction) to get the signal from the noise that's hitting your phone constantly. This is loosely the reason for why it drains the battery (even more so during cold starts).

I started to build a gps tracker for my cat which wouldn't require a monthly subscription- after burning the first micro-controller I gave up and decided to leash train my cat. Now my cat is leash trained.