Short answers, pulled from the story.
Bounty hunters in the United States derive their authority primarily from the 1873 Supreme Court decision Taylor v. Taintor and from the bail bond contract a defendant signs. Under this framework, they may enter a fugitive's legal residence without a warrant, cross state lines in pursuit, and make arrests without due process, because the bail agreement is treated as a civil contract rather than a state action.
Bounty hunters are typically paid a commission of around 10% of the total bail amount owed by the fugitive. The exact commission varies case by case depending on the difficulty of the assignment. If the fugitive escapes and the bail is forfeited, the loss falls on the bondsman, not the bounty hunter.
As of 2008, Illinois, Kentucky, Oregon, and Wisconsin had banned commercial bail bonds and prohibited bounty hunting. As of 2012, Nebraska and Maine similarly prohibited surety bail bonds. Texas and California require a license to engage in bounty hunting.
Duane Chapman was arrested in Mexico after apprehending Andrew Luster, a multi-millionaire rapist and fugitive. Chapman was subsequently released but was later declared a fugitive by a Mexican prosecutor and arrested in the United States to face extradition. All charges were eventually dropped after the statute of limitations expired.
As of 2003, bounty hunters claimed to catch 31,500 bail jumpers per year, which the industry described as roughly 90% of all people who skip bail.
Bounty hunters have no legal authority to arrest anyone outside the United States. In most other countries, taking a person into custody without authorization can be charged as kidnapping. Daniel Kear was extradited to Canada in 1983 and convicted of kidnapping after abducting a fugitive from Canada to return him to Florida.