Does the skin effect take place on both surfaces of a pipe conductor, outside and inside, or only the outside? (I don't understand the underlying physics)
A tube of current does not produce a magnetic field on the inside thus, skin effect (and the majority of current) exists at the outer surface.
It's a similar story when you analyse a length of coaxial cable; the magnetic field external to the cable caused by forward current in the inner conductor is completely cancelled by the return current flowing back through the shield (aka tube). Thus there is no net external magnetic field (as we know and love about coaxial cable). But, there is an internal magnetic field between inner conductor and shield however, that field is purely due to the forward current carried by the inner conductor.
This gives rise to the fact that for an ideal length of coaxial cable, the shield (aka tube) possesses zero inductance. It can only have zero inductance if there is no net magnetic field associated with it. And, of course that means there is no internal magnetic field produced by the shield.
A self-made QuickField simulation might help you believe this: -
The above image is for coaxial cable but, the inner conductor is not being used to pass current. The field inside the tube/shield is effectively zero despite the shield passing 1000 amps. This means there is no skin effect on the inside of a tube carrying current. It's a physics thing really but, is a really important aspect of how coaxial cable works.
For a sheet carrying current, skin effect occurs on both sides of the sheet.
would that mean that a sheet metal piece with, let's call it "top", surface equals that of the outer surface of a half pipe, and the "bottom" surface of the sheet likewise equals the other half pipe's outer surface, one could get away with half the material when using the sheet metal vs. the pipe, yielding the same efficiency?
Whether you could get away with it depends on your application. The beauty about the tube/pipe is that it produces an equal magnetic field in all directions along the length of the pipe. This may indeed be superior in some applications compared to a sheet antenna.
Also, the inductance of a sheet will be smaller hence, the overall magnetic field produced for a given current will be less than that of a tube (because it's inductance is higher). This is down to what inductance means: the amount of magnetic field produced (per turn) for a given current (per amp).