We recently reported the discovery via molecular-dynamic simulations that single-walled carbon nanotubes (SWCNTs) with different diameters, lengths, and chiralities can coaxially self-assemble into multi-walled carbon nanotubes (MWCNTs) in water via the spontaneous insertion of smaller tubes into larger ones. Here, we extend that study to investigate the various water structures formed between two selected SWCNTs after such coaxial assembly. Depending on the tube geometry, typical water structures, besides the bulk phase, include a one-dimensional (1D) ordered water chain inside the smaller tube, a uniform or nonuniform water shell between the two tubes, and a "boundary layer" of water near the exterior wall of the larger tube. It was found that a concentric water shell consisting of up to three layers of water molecules can form between the two SWCNTs, which leads to a class of carbon-water-carbon composite nanotubes. Analysis of the potential energy of the SWCNT-water system indicated that the composite nanotubes are stabilized by both the tube-tube and tube-water van der Waals interactions. Geometrically confined between the two SWCNTs, water mono- and bilayers are found to be stable, highly condensed, and ordered, although the average number of hydrogen bonds per water molecule is reduced. In contrast, a water trilayer between the two CNTs can be easily disrupted by thermal fluctuations.