In the films
In Blade Runner Deckard tells Gaff that he’s retired, which suggest he was formerly on some kind of contract, otherwise he might instead have said something more along the lines of ‘I’m not in that line of work any more.’
I don’t know how reliable the Blade Runner wiki is but the Blade Runner (police) entry states:
Blade Runner is the term used to refer to members of the "Retirement
Division" of the LAPD. The main purpose of a Blade Runner is to
eliminate replicants.
Members of police departments are typically employees, but they could be part-time. Although I can’t find any explicit reference in the films to a blade runner being a full-time job, it doesn’t make sense for a state department (the police in this case) to train someone as a killer and then leave them unemployed because unemployed assassins tend to turn to the private sector for an income. They would at the very least be on some kind of retainer or contract to keep them in check when there aren’t replicants running around the place.
In the source novel
Blade Runner is loosely based on Do Androids Dream of Electric Sheep? and that indicates that Deckard is a full-time member of the police. The bounty seems to be a bonus.
Pris stays the most bounty hunters are on a contract:
‘What’s a bounty hunter?’
‘That’s right. You people aren’t supposed
to know. A bounty hunter is a professional murderer who’s given a list
of those he’s supposed to kill. He’s paid a sum — a thousand dollars
is the going rate, I understand — for each he gets. Usually he has a
contract with a city so he draws a salary as well. But they keep that
low so he’ll have an incentive.’
Do Androids Dream of Electric Sheep? Chapter 13
She could be lying or misinformed, but the novel starts with Deckard getting up and arguing with his wife about her use of the Penfield mood organ. Then after breakfast he checks up on his electric sheep, and chats with a neighbour about wanting a real animal (bold added, italic in the original):
‘God,’ Rick [Deckard] said futilely, and gestured empty-handed. ‘I
want to have an animal; I keep trying to buy one. But on my salary, on what a city employee makes —’ If, he thought, I could get
luck in my work again. As I did two years ago when I managed to bag
four andys during one month. If I had known then, he thought, that
Groucho was going to die … but that had been before the tetanus.
Before the two-inch piece of broken, hypodermic-like bailing wire.
Do Androids Dream of Electric Sheep? Chapter 1
He finally arrives late to work (having stopped to window shop at a pet store) and is only then told about (bounty hunter) Holden having been shot, but he still doesn’t know about the case, so Deckard’s not been specially summoned with an urgent call to arms. It comes across as the beginning of an ordinary day at the office. He also has his own desk:
Rick [Deckard] said, ‘I’ll find out from Harry Bryant.’ He felt
irritable; office gossip annoyed him because it always proved better
than the truth. Seating himself at his desk he pointedly fished
about in a drawer until Miss Marsten, perceiving the hint, departed.
Do Androids Dream of Electric Sheep? Chapter 3
