In every case I have read about, it is lawmen or detectives, not the independent killers you see in films, who seek out and collect these rewards. Then again, saying that you would be willing to ride out after a criminal was sometimes enough to get you a badge and to give you the auspices of the law.

As an example of a lawman collecting a bounty, there's the famous case of Billy the Kid. Pat Garrett, the sheriff of Lincoln County, was tasked with capturing Billy the Kid. The governor of the state, Lew Wallace, had set a $500 reward for Billy's capture. Unlike the movies, this reward came out in a newspaper ad:

BILLY THE KID

$500 Reward

I will pay $500 reward to any person or persons
who will capture William Bonny, alias The Kid, and
deliver him to any sheriff of New Mexico.
Satisfactory proofs of identity will be required.
LEW WALLACE, NEW MEXICO

(Source: The West of Billy the Kid, by Frederick Nolan)

This phrasing was problematic: did it imply that the sheriff himself couldn't collect? was it incentive for someone else to bring Garrett proof? Garrett clearly didn't think so, and claimed the reward himself when he killed Billy.

Interestingly, the state originally tried to back out when Garrett tried to collect the reward for killing Billy. The new governor claimed that Wallace had made a "personal offer" that was not binding on the territory (this was a barefaced lie). It was only through Garrett's insistence and lobbying that the territory finally paid him on February 18, 1882, seven months after the event. By that time, private citizens, "disappointed by the territory's lack of fiscal integrity," had sent Garrett $1,150. Taken from Pat Garrett: The Story of a Western Lawman, by Leon C. Metz.

It wasn't just the state who would set bounties on people; companies, especially those who had been robbed or who had employees killed, would set private bounties on top of what was already being offered. Also, private citizens would contract those who could lead to their stolen property being returned.

Apart from rewards offered for dead Indians, I know of no case in which the "dead or alive" referred to killing a criminal in cold blood and then receiving a reward for it. In the cases I've read about, it's usually stated that the reward will still be given if the fugitive is killed in self-defense (often, cash rewards were only offered if the fugitive was taken alive and stood trial). For example, when Robert Ford famously gunned down Jesse James (while the latter had his back turned, adjusting a picture frame), Robert and his brother Frank hurried to collect the reward. Instead, they got a murder charge for killing James in cold blood. Though convicted, they were pardoned almost immediately.

The Pinkerton National Detective Agency made good money collecting rewards offered for everything from murderers to robbers, cheats, swindlers, and cattle rustlers. They were semi-official, as the agency received massive government funding, and the same agents could be hired by the government or by private citizens. The men and women whom Pinkerton employed were themselves sometimes on the wrong side of the law. Such lives make for fascinating reading as they swing between collecting rewards for bringing criminals to justice to themselves committing criminal acts. There was often a hazy line between murder-for-hire and pursuing a reward without looking too hard at who was offering it.

I think you might be interested in the story of Tom Horn. He was a scout, a Pinkerton detective, an outlaw, and a hired gun for unscrupulous cattle companies. You can read his own autobiography, written to vindicate himself in the murder of a 14-year-old boy he was accused of murdering for $500. It is heavily biased, of course, but makes for a fascinating first-hand account. It is in the public domain, published in 1904, shortly after his death by hanging: https://archive.org/details/lifeoftomhorngov00hornrich A modern book about his life is Tom Horn: Blood on the Moon, by Chip Carlson, which I recommend (by the way, Carlson comes to the conclusion that Horn was not guilty of that murder; he also runs the tom-horn.com website).

Here's how Horn describes working for the Pinkertons: "My work for them was not the kind that exactly suited my disposition; too tame for me. There were a good many instructions and a good deal of talk given to the operative regarding the things to do and the things that had been done." He tells the story of tracking down some train robbers, following leads across many miles and capturing another fugitive along the way. Interestingly, it's once the sheriffs and posses give up that the Pinkertons receive word from their boss "to go at 'em," meaning to start the detective work.

Another former Pinkerton, Charles "Charlie" Siringo, wrote Pinkerton's Private Detective, which the company blocked him from publishing (he ended up changing some details and publishing it with the Pinkerton name removed). To get revenge on the company, he published Two Evil Isms: Pinkertonism and Anarchism. The book is a fascinating tell-all about the Pinkertons, their methods, and their corruption (obviously written by a disgruntled employee, so read it with an eye to that). Among his claims are that the Pinkertons hired themselves out without any intention of catching criminals in some cases, only reaping per diem payments and the fees for associated costs. He also claims that, even when rewards totaled in the tens of thousands of dollars, the Pinkertons were bought off not to pursue the reward. Here's an interesting quote from that book:

"With all of the agency's faults, I must confess that they do a lot of good work in running down crime for money. If they did not, they could not keep their heads above the dirty water in which they constantly flounder."

Here's a link to the book: https://archive.org/details/cu31924002532475

(I've never tried contributing a top-level comment on this subreddit before, and I know there are strict rules about what counts as quality content, so I've tried to offer lots of sources and explanation, and to note gaps in my own research. I, of course, welcome other contributors to fill in the gaps or to note where I've made an incorrect supposition.)

Edit: Quick edit to add line breaks to Billy the Kid's wanted ad, since I had used single-line breaks, which ended up in one big text block.