For keyless entry, cars are always transmitting. They look for a response of a key. In case of my car, if a keys comes close enough, it lights up the doors and doorhandles.
Some older fobs do recharge when in the car’s ignition. My old bmw did this and it was recommended to swap the keys from time to time to make sure they didn’t run out off battery. I guess this can’t be the case with the “keyless” keys though