People are funding client skeptics, and people are finding Climate Change studies.
You're right in that we have two groups - but only one is involved in actually science.
When you receive funding only when your "research" produces the desired results it becomes nearly impossible to have unbiased results. It becomes propaganda masquerading as research. To actually perform real research, the researcher must receive funding regardless of result.
The problem with the skeptics is that their "research", which is always biased, is taking away from the real research that is being done. When an outsider observes two publications making opposite claims, both publications are discredited. And if you ask that outsider which publication they believe, they will usually pick the one they want to be right - which is the one that says they can keep on burning oil.
The scientific community knows that climate change is real and that human activity is to blame. But the general populous does not partly because of the fake research and the arguments it spawns. So no, we shouldn't accept funding from all sides. Funding should only come from a neutral side - if the rich want to fund more they can donate funds to that neutral side.