Full scholarship and student advisor relationship If a phd student is completely funded by an outside institution and don't cost the school or the university anything, what impact would that have on the student relationship with his advisor? Especially on things like interest on the student progress.
The main effect of external support on a student/advisor relationship is that there is less external pressure for production of short-term results.  Whether this is a good thing or a bad thing depends strongly on the individual student and advisor.  


With less pressure on the student, there is more freedom to develop a unique research agenda but also more opportunity to get "lost" in the possibilities and waste time.
With less pressure on the advisor, there are fewer boss/employee dynamics in the relationship and generally less strain, but also more opportunity for the advisor to be distracted by other short-term concerns.


A secondary effect is that the student may have less opportunities to be apprenticed in grant-writing, since their support is already assured.

In sum: with a good advisor and a student with a mature outlook, the freedom of external support is a blessing.  If either of those two ingredients is lacking, it can be a problem to lack the forcing functions of external support.