With my previous coach, I picked and cut the music, she choreographed my freestyle programs, and I choreographed my showcase programs. With my current coach, I pick and cut music for everything. She choreographed my freestyle program and my last showcase, and I went to her with an idea for my showcase. She did about half the choreo so far but I may wind up finishing it. I tweak almost everything and sometimes she tweaks it right back (i.e., when I make it easier...)
When I choregraph for my adult students, they pick the music and I choreograph. Sometimes they have an idea of what they want to do and I choreograph around it. With kid students, I ask if they have any particular music they want and if not, I pick and cut the music, then choreograph.