There are (at least) two ways of solving this:

Either increase the so-called SpherePoints to a large number:

Graphics3D[{Red, Sphere[{0, 0, 0}, 1], Yellow, Sphere[{0, 0, 0.001}, 1]}, Method -> {"SpherePoints" -> 1000}]

Or (better I think), create a custom function that plots colored-half-spheres:

ClearAll[ColoredHalfSpheres]
ColoredHalfSpheres[p:{x_,y_,z_},r_,col:{color1_,color2_}]:=
Translate[First@SphericalPlot3D[{r,-r},{\[Theta],0,Pi/2},{\[Phi],0,2Pi},Mesh->None,PlotStyle->col,Axes->False,Boxed->False],p]

Graphics3D[ColoredHalfSpheres[{1, 2, 3}, 0.4, {Red, Blue}], Axes -> True]

Note that the first solution is very memory/graphics intense because it creates many many triangles to represent the sphere. The second might take a bit longer to create but then is much smoother.