There's very little research on "natural" products especially in the USA, mostly because there doesn't have to be. "Artificial" products have to be proven safe before they're allowed on the market, but "natural" products are assumed safe until proven otherwise (and have to be proven extremely dangerous to be banned).
That doesn't mean that all natural products are less safe than artificial ones, but it also doesn't mean that the potential isn't there. Hemlock is natural too.
What disturbs me about stevia is that the little research that is available raises possible concerns. Laboratory studies of stevia have found potential cancer and reproductive-health problems in lab animals. They may not affect humans, but then again, they might.
While it's true that stevia has been used in it's natural form for thousands of years without apparent ill-effect, that doesn't mean there aren't any risks, especially when used daily in a concentrated form, or in the amounts that Americans tend to use sweeteners.
I think this article summarizes the issues pretty well
Personally, I'm more assured by the research on artificial sweeteners than the research (and worse the lack of research) on stevia.
I used to use quite a lot of artificial sweetener, but I've been progressively cutting back. A bag of Splenda that used to last me two months, now lasts so long I can't even remember when I bought it (I should probably date it with a sharpie when I buy it, to keep track).
I also drink too much sugar free soda (though I'm more concerned with the acid's affect on my teeth than the effects of the sweeteners), so I've been diluting my soda with water or tea.
I've found that sweeteners and sugars (natural and otherwise) are like salt, the less I eat the more sensitive I am to the flavor. For example, apples that I used to consider far more tart than sweet, taste sweeter to me now.
Low-carb eating in general has recalibrated my taste buds (even though I'm not eating as low as I could and should be).