Truthfully? We don't know.
It seems safe to assume that after the events of Captain America: The Winter Soldier, SHIELD are no longer paying Steve Rogers for his services, and by the time Avengers: Age of Ultron occurs, Steve is a full-time Avenger, leading a team funded by one of the richest men on the planet, Tony Stark.
However, I don't imagine for a second that any of The Avengers take a paycheck from what they do, for two reasons.
It is mentioned in Avengers: Age of Ultron that Tony Stark sends in a relief team to areas that have been affected by Avenger activity. This makes it sound as if the Avengers are operating as a (global) public service, one that seeks out threats and dispatches them as they see fit, accountable to no-one but themselves and ensuring that the mess they leave is cleared up too. That implies a lack of government backing (further supported by Tony Stark's claims that "he's just the guy that pays for everything").
From a character perspective, the Avengers are clearing up messes that they helped create. I've gone into this area (specifically concerning Thor) more here, but it seems safe to say that the rest of the Avengers would follow a similar thought pattern - that they have a responsibility to help, with or without pay.
Add in the fact that the team leader of the Avengers, Steve Rogers, specifically states that he can't afford a property in Brooklyn, and it seems safe to say that members of the Avengers avenge for free.