Should all the departments work independently or should be there be a systematic approach to work...where all the departments work together?

Basically all the department works Independently, because there roles and responsibility are very clearly defined  but  automatically they are bind in a system which they have to follow for the smooth working .. solely no department runs perfectly its the collective efforts and collaborative system approach which makes the organization successful.

It all depend upon organization structure.In a matrix organization all departments work dependent.Even in the professional firms department work interrelated with standard sections.However integration reduce communication,coordination gaps

Every department has its own and exclusive responsibilities and should work independently, otherwise there will be no fun in creating departments.

YES there should be a defined system to generate a harmony among them to work more effectively and more efficiently.