Does a structural engineer design buildings or does the architect solely do that?
Usually, the architect works with the owner/developer to provide drawings that include visual concepts (how the structure looks), layout (room sizes, bathroom & stair locations, etc), and other things like that. The structural engineer is usually hired by the architect to provide drawings and calculations for the "skeleton" of the building (or how the building will be able to stand after gravity, wind and seismic loads are applied to them). They will specify what sizes, material, and spacing of the beams, studs, columns and joists. They will also specify concrete slab, wall, and footing sizes, and other things like that. The architect, structural engineer, and mechanical engineer (designs the heating, air-conditioning, etc) work together to coordinate a drawing package that is submitted to the city for a permit.
Overseeing construction can be what a structural engineer does to a small extent, as with a design-build situation, but it can also be what a Construction Manager does (on large projects, say a hospital or an office building) or just the contractor (on a small project like a house). That said, most U.S. civil engineering programs that train structural engineers have a class or two (if not a whole concentration) in construction management. Designing buildings and making sure they’re safe is exactly what structural engineers do, though. It’s design of buildings (and large non-buildings, such as bridges and dams) from a functional and pragmatic standpoint, where the concerns are things like constructibility (i.e., “Can it actually be put together at an affordable cost?”) and load resistance (e.g., “Will the roof support the weight of the maximum amount of snow that it’s reasonable to expect will accumulate during a harsh winter?”). Design where the concerns are aesthetic and philosophical
