Same thing happened to me. I got a price from an auto body shop, showed the appraiser my estimate when he came to check my rig out, and he approved the cost. Cut me a check on the spot (minus my deductible). I actually ordered a v-drop bumper that didn't cost as much as he wrote the check for and I kept the difference.
Of course, they got me when I renewed my insurance this year, my rates when up (year one of three that I'll be paying in surcharge for that little fender-bender). As per the below, depends on your insurance co.
(p. s. ) mine didn't ask for pictures or to see the fix after they paid me off.
zman