I have built two concentric hemispherical shells from two spheres centered at (x,y,z) = (0,0,0), and added a thickness to the outside shell to fill the gap between the two shells. However, I have found that -- despite the shells being of the same proportionality (height = 1/2 diameter) -- I do not get a perfect intersection of the inner wall of the green shell and the wall of the blue shell, as illustrated below.
You can join the two meshes in object mode. Then in edit mode you you can select the two edge and perform a Bridge Edge Loop. That will also serve as a check to make sure you have the same number of vertices.
If you want two separate objects. You could start with one hemisphere, [solidify modifier], apply the modifier, The select the inner portion and menu [separate] to form two separate meshes.