We say that the magnitude of the electric field inside a conductor is zero, whether a hollow or solid. However, at certain charge densities, the magnitude of the electric field inside the conductor comes out to be not zero. How can both these things right?
If you are talking about a conductor object whose Gaussian surface is in such a shape that due to symmetry, electric field vectors of point charges on the conductor cancel each other out, then, yes, the magnitude of the electric field would be zero inside that conductor.
No charge density would change this property because, symmetry would always be there. However, if you somehow change the charge distribution of the conductor, then symmetry would be disrupted and the magnitude of the electric field due to the charges on the conductor wouldn't be zero. This can be done by applying an external electric field through the conductor; however, although the magnitude of the electric field due to charges on the conductor is not zero, the external electric field would cancel that electric field. Therefore, the electric field inside that symmetrical conducting object would still remain zero. This device is called a "Faraday Cage" and commonly used in devices we use everyday such as for protecting the electronic components of a mobile phone from external electric fields.
If the conducting object is doesn't have such a proper symmetrical shape, then the electric field naturally wouldn't be zero inside that conductor.
Inside a conductor there can be no permanent, static electric field.
If there were , electrons ( or sodium and chloride ions in the case of brine ) would move to opposite sides ( under the influence of the external electric field ), creating their own electric field, exactly cancelling out the externally applied electric field.
The net electric field inside a charged conductor is zero. Because then it is at equilibrium and the charges position themselves so to minimize the repulsive form between them. So they end up on the surface of the conductor. All the charges on a charged conductor resides on its surface and create an electric field perpendicular to the surface. When you mentioned charge density I figured that you were talking about the case where a charges conductor is inside another one. As in this picture. You can see the electric field between the surface of the outer conductor and that of the inner conductor and an electric field of zero inside the negatively charged conductor. It is here that your problem might give you the charge density inside the positively charged conductor.