To what extent should a professor give students freedom to do independent research?
At the end of a PhD a student should be able to conduct their own research.  To reach this goal it is helpful to reduce the amount of direction given during the course of a PhD.  When the student starts, some defined goals and guidance can allow the student to make rapid progress.  
Students should make unique contributions and understand a broad range of concepts.  For example, producing a PhD thesis just on one tiny component of an analysis is a bad idea.  Students should be allowed to make mistakes, but be prevented from loosing a large amount of time on bad choices.  
Within highly competitive research environments one might have to defend a students work with additional members of staff, since students are typically slower than experienced post-docs.
All students are different.  Therefore, the level of guidance needed should be balanced accordingly.
The final mark for the PhD should reflect the ability of the student.





  What do you mean by independent research?


Completely independent research implies that one has to define the project and direction, using knowledge of the field of study.  One needs to read journals and join workshops/conferences to determine what has not been studied and can be studied within a reasonable time frame.  Given this, one then writes software/develops an experiment to measure the chosen parameter(s). (There are many holes that a student can fall into along the way). It is a lot easier to carry out independent research as a post-doc or professor, since one builds up knowledge.


  Should your students conduct independent research and publication in the field of their PhD/MSc research topic or it can be in any other field of their major and interest?


Their PhD is typically on one topic e.g. particle physics. The independent research should happen within the discipline that is being assessed.


  If their independent research is not in the field near to their PhD research, should they inform you about their other research projects or not?


PhD theses contain a fraction of a student's work. They do not contain all of the work of a student. For example, one might build a silicon detector for two years and then spend 1.5 years performing a data analysis. The silicon detector work (even if performed independently) may not end up in the thesis. It is very hard for examiners to assess work that is not in the thesis.

(From ~13 yrs of supervising particle physics PhD students.)