What makes a lens a lens? I am asking about the way that a convex lens works.
All the lenses I have ever seen were solid glass or solid plastic.
If I have a convex sheet of clear glass or of clear plastic, shaped into a parabolic curve similar to the parabolic curve of a solid lens, will it have the same ability to focus light as the solid lens would?
 A: No, it wouldn't.  
If you have a normal sheet of plastic, then even when curved, the two sides are very nearly parallel.   When a light ray enters and then exits an optical medium with parallel sides, the incoming and outgoing rays are also parallel.  The ray may deviate, but it does not change direction.  Since the direction is unchanged, there is no lensing.

Source www.bbc.co.uk
But when the exit surface is not parallel (as in a triangular prism or a traditional lens shape), then the exit direction for the ray may be very different from the incoming direction.  This change of direction is crucial to the lensing.  In this image the entry and exit points for the ray are not parallel, and the final ray is not parallel to the original.

Source: wikimedia
While you can't take a regular (flat) sheet of plastic and lens with it, you can take a regular lens and chop it up into lots of separate thin lenses.  If you put them together, you can get a sheet that lenses.  It just doesn't do so as clearly as a smooth lens.  This is a Fresnel lens

Source: wikimedia
A: The short answer is: what counts is the difference of orientation of the air/material interface (or any 2-materials interface) between entry and out point of the ray.


*

*For a simple shell the 2 interfaces are pretty parallel: bad lens since one just undo the refraction tilt of the other. 

*An hollow "lens" with 2 curved shells is indeed just a couple of such 2 parallel interfaces, so it's not better. 

*A usual lens is solid, so the 2 sides of the material have different orientation (i.e., surface normal), so Snell-Descarte law at the out will not undo the entry Snell-Descarte transform in the original direction.

*As pointed out in the long answer, thickness does not matter: only 2-sides orientation + being solid. Thus the non-intuitive functional saw-profile of the Fresnel lens.


BTW you know more than glass and plastic lenses: you know water lens (waves), your eye, hot air layers (mirages) ;-) 
