Improvisation and ownership

I had a long discussion with a choreographer friend about caveats of the spectrum of composer vs. performer musical contribution to a piece. Dance is typically unwritten and requires choreographer coaching from start to finish; music in the classical tradition is typically scored so the composer's presence is superfluous until the final stages of rehearsal, and is never required. But even a through-composed piece requires the performer to put herself into the piece; one that requires greater degrees of improvisation is only asking a certain kind of performer, i.e. one prepared to improvise, to put more of himself into the piece.

So what determines whether a piece is the composer's, and how do individual performers define and enforce their thresholds for their original contribution? These answers inform how I instruct performers to improvise.