There is no such thing as a GPS radio in a phone in the sense of something that broadcasts a signal. GPS signals are broadcast by satellites and any device on earth can choose to listen for the signal and triangulate their position by receiving the signal from multiple satellites. GPS doesn't send out any signal from your phone to work. In the same way that your eyeballs don't need to emit light for you to see (unless you're a superhero). Phones broadcast signal to connect to cell networks and wifi networks.

Now if you expect your phone to show your position on a map (using the GPS coordinates it calculated from the satellite signal) then it likely needs to retrieve map data from the internet (and in doing so it might be broadcasting over cell or wifi frequencies). But that's a different story.

Much confusion is caused by people saying "I tracked X with GPS" with the implication that GPS allows tracking of remote objects. It doesn't. It allows the remote object to know its own location. For that information to get to someone else (the spy in the chasing vehicle) the remote object needs to be broadcasting that location information through some other means (which has nothing to do with GPS).