When I saw them in the bottle, I thought they would be more different than they actually are.
On the nail, these are very similar, with Private Dancer being a little more reflective (it has more of a chrome finish that Cheyne Walk). It also is a little brighter and it seems that Cheyne Walk has more red in the base. Cheyne Walk also has more shimmer particles.
All of that said, my husband couldn't tell these apart. He stared at them for 2 minutes and couldn't say how they were different. So, for those people that aren't "nail polish people" these probably could pass for dupes. The formula on both was very similar, as it took 3 coats to reach opacity on both.
Personally, I think they are very close, and unless you just have the need for both, I would say you would be okay with one or the other. Nails Inc Cheyne Walk retails for $9.50 a bottle and Deborah Lippmann Private Dancer is $18. The Lippmann is almost twice as much, but the Nails Inc is a 0.33 oz bottle and the Lippmann is 0.50 fl oz.