This is a big and if we are honest a philosophical question because there is no absolute necessity nowadays to hunt pumas (aka mountain lions). This activity, pastime or form of human entertainment (primarily) is carried out because some humans like to do it. Their desire to hunt probably originates in the ancient human lifestyle of hunter-gatherers of 12,000 years ago. This ancient hunting culture remains in the DNA of many people, mainly the male human. Most humans have lost it and many regard hunting as repugnant because it is no longer required for food but is carried out for pleasure.
Pumas are hunted with dogs in North America where it is legal in many states to hunt them in this way but you’ll have to check on a state basis for the law on hunting this cat. There are pictures, online, of pumas in trees trying to escape the dogs.
Pleasure hunting aka sport hunting is also big business. Businesses will always spring up to make money out of activities that a significant percentage of the population like to do and there is a significant percentage of Americans who like to kill animals for fun. So one reason why pumas are hunted is because financial profit is part of the fabric of the activity. When that happens it is hard to stop it. Business owners are very reluctant to give up their businesses. And they lobby governments to protect their source of income.
In 1990, 1,875 pumas were hunted to death in eleven western states. A major reason today why puma hunting is still allowed is because it seems that the population size of pumas is stable. However, the entire population of pumas in the eastern half of America has been eradicated, hunted to extinction combined with loss of habitat and the other reasons such as retaliatory killings by farmers.
As at 2002, hunting pumas was prohibited in most South and Central American countries but cats that attack livestock are often shot. Also as at 2002 the puma was not protected in Ecuador, El Salvador and Guyana. Now, puma hunting is regulated in many countries such as Canada, Mexico and Peru.
So the authorities have decided upon a compromise between banning puma hunting and allowing free rein for the shooters. This is probably because, overall, the puma numbers are decreasing and therefore there is a concern for their conservation although the species is classified as ‘Least Concern’ by the authority charged with deciding these things; the IUCN Red List. This classification on the puma’s ability to survive in the human world opens the door to legislation which permits puma hunting. If the Red List classified the puma as ‘Endangered’ or ‘Critically Endangered’ or even ‘Vulnerable’ state governments would have to stop or severely restrict mountain lion hunting.
For this reason you’ll find that the sport hunting associations and hunting businesses lobby the IUCN people to ensure that species which are hunted are classified in such a way which allows them continue to hunt them.
Conflict with Humans
There was mass, legalised hunting of pumas in the early part of the 20th century following the introduction of sheep and cattle to North America. In 1937 the Animal Control Act was passed (ADC program). This act is still in force. It contributed to the killing (I guess mainly by shooting) of 66,665 mountain lions between 1907 and 1978. In 1990 41 pumas were destroyed under the ADC program.
Although an enormous amount of money has been spent eradicating livestock-killing pumas, the number of livestock killed is insignificant compared with the overall population size.
Problem pumas, those that have become desensitised to humans because humans have built settlements on their territory are still sometimes hunted and shot dead because they are a threat to people. However, I have noticed that some authorities prefer to tranquillise and relocate but that can go wrong (too much tranquilliser kills the cat) and relocation can be tricky and unviable. Conflict with humans is a significant reason why pumas are still hunted to death. This is not hunting as a pastime but in the interests of human welfare. But the underlying reason for it is rooted in human population growth squeezing out other species.
Note 1: The references to numbers and information other than my own comes from Wild Cats Of The World ISBN 0-226-77999-8.
Note 2: I am an animal lover. I hate the hunting of all animals which is why this article, although based in fact, is biased.