My understanding of this from my aeroeng degree and advanced aerodynamics is that the winglet generates thrust, by extracting energy from the vortex flow at the wingtips. However, this is only so if the winglet has been correctly designed taking an accurate flow pattern into account. Vis a Vis newtons laws are not destroyed; energy is neither created or destroyed. The doubters are also partially correct in my (humble) opinion in that winglets are used to increase Efficiency as opposed to generate thrust, as they also provide a little lift, thus reducing the required incidence and thus marginally reduce induded drag in the cruise. Furthermore, (this is where it gets difficult) the flow triangle of forces taking into account of the extra drag of the winglet actually REDUCES the apparent AOA, thus leading to a further reduction in drag. I believe the savings are less than 1% of operating costs; when translated into total fuel costs for an airline per year however, this saving could run in the tens of millions.
PS, the winglets should also reduce the vorticity on landing and thus reduce any wake turbulence, however, I believe this to be a secondary reason for installing winglets.
By the way, anyone wanting the official line should check out
http://www.fluent.com/solutions/articles/ja133.pdf
which concisely settles any argument.