They still exist. Here, for example, is a recent article from "The Atlantic" about present-day bounty hunters.

http://www.theatlantic.com/politics/archive/2015/07/does-the-bounty-hunting-industry-need-reform/399224/

You can find state laws regulating bounty hunters by looking in states' legal codes/statutes. E.g., California Penal Code sec. 1299 and Insurance Code sec. 1810.