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.