Your interesting question has two components that need to be addressed. Based on the first part of your question, I assume you are referring to people who do not have conventional religious belief systems, but I'm not sure if you are restricting the discussion to Christianity or to the larger world of religion, which includes all established religions. The second part--what constitutes a moral compass and who might have one--is a bit easier to handle.
We first must address the characterization of "non-believers, doubters, or religious skeptics," which leads to an important question: non-believers and skeptics of what? If your question is based on the assumption that there is one true religious belief system, I would simply point to the billions of people on the earth today who follow different religions but believe that their belief system is the only true belief system. All can't be right--or can they? Unfortunately, we can argue till the end of time whether there is one true religious belief system--and, from a logic standpoint, we would be guessing because religious beliefs are based on faith, not knowledge, and arguments about religion are not resolvable through the exercise of reason alone. Even more important, no one can "know" which religion is the true religion except through faith, and faith, I still argue, is not knowledge. To put it simply, faith is faith, and knowledge, knowledge, and never the twain shall meet, at least not anytime soon.
The question of what forms a moral compass is more readily answerable. If a moral compass can be defined as a set of beliefs that allows a person to understand what actions can be good for or hurt other people--and I realize that this is a simple definition--we can look at history to determine whether a moral compass is the result of a single religious belief system. From the earliest historical periods, there is evidence that mankind has attempted to codify what is good and bad in human behavior. If we look at the Code of Ur-Nammu (2000 B. C.), the Code of Hammurabi (1754 B. C.), Hittite Laws, Mosaic Law, or more modern codes like Napoleonic Law and English Common Law, mankind has been attempting to set forth moral and civil law compasses from the beginning of recorded civilization. And although many of these codes were supported by the religious beliefs prevailing in a specific culture, the codes encouraged what we would call simply ethical behavior--for example, don't lie, don't steal, don't kill (still working on this one in a big way), leave your neighbor alone; don't cheat. Religion, in some of these cultures, helped frame the codes but was not the reason these codes were enacted.
In short, then, there are many doubters, skeptics, and non-religious people everywhere we turn in the world or next door, but these people may, and often do, have a moral compass that leads them, if not to do good, then at least not to do any harm. Conversely, history has given us uncountable examples of people with strong religious beliefs whose "moral compass" is pointing directly south. A reasonable conclusion to this issue is that moral or ethical behavior is, as often as not, independent of a religious belief system.
One's morality is not determined by a person's religion. Even if a person claims to be a non-believer, a doubter or a religious skeptic; this does not mean that he is immoral. Yes, religion can be a factor in shaping one's morality, but they are indeed other variables to consider. It can be the influence by friends, co-workers, parents, teachers, different types of mass media, interactions in social networking sites, and many others.