If I recall correctly, the person who bought the insurance would file the claim. That would usually be the shipper. I had to file a claim, but it was a decade ago. Things may have changed.
EDIT: This made me wonder, so I went to the USPS website and found the following:
How to File a Claim
Either the person who sent the mailpiece or the person who received it may file a claim for insured mail that is lost, arrived damaged, or was missing contents. The person filing must have the original mailing receipt. Each claim must be filed within a certain time period and include proof of insurance, value, and damage.