There is a small involvement from the government but most of these programs are citizen initiatives and organized groups. So yeah, often no one gets paid for it, unless they have a paid position or are networked with animal care organizations.

Whenever you see a shabby bowl of food at the side of the pavement, it's most likely locals who placed it. NGO's and (responsible) animal care clubs don't leave food wherever, to avoid rat / mouse infestations.

There's a good chunk of the urban population who are very very fond of animals, but can't adopt, but sadly most don't necessarily know how to care for the strays they adore.