Legally it's expected for you to. BUT, and this may well be echo'd - if you tell the insurance company you've had a prang, the other person has paid for the work and you DONT want to claim, your insurance next year goes up - as they see it that you've had an accident!
People have prangs and settle it without telling Mr Insurance, day in, day out. That alone keeps the insurance costs lower than they could be.
If you are happy to get her to pay the bumper and tailgate - and have a NON-WROTE OFF Clio that's cosmetically repaired and does the job - Do it. But when coming to sell it - the boot may be spotted and turn off an interested party.
Going through insurance - getting it wrote off, getting a huge lump sum back, repairing for cheap and driving a CAT D car around the the boot and bumper done + cash in the bank = other option. This option will see your insurance rise a little though.
Food for thought.