In the Einstein model, atoms are treated as independent oscillators. The Debye model on the other hand, treats atoms as coupled oscillators vibrating collectively. However, the collective modes are regarded here as independent. What is the meaning of this independence and how does it contrast with the Einstein model?
The independence of the oscillators is a consequence of the assumed harmonicity of the oscillators, i.e. the forces being linear functions of the displacements. Once anharmonic terms are considered, these will give rise to scattering of oscillations on each other which is among others one reason for the limited thermal conductivity of solids.
you mean in the Einstein model the independence is a consequence of assumed harmonicity of oscillators? Or in the Debye model the collective modes are regarded as independent because of the assumed harmonicity?
In both models. Also note that the Debye model reduces to the Einstein model when the energy bands are flat as then you can construct localized oscillations from the degenerate oscillators. Often, the bands corresponding to optical phonons are not very curved so that they can be approximated by Einstein oscillators.