In my limited experience the seller handles shipping. Sometimes it's been great packaging and sometimes not. Once the seller decided insurance was too much and it shipped uninsured. Fortunately it turned out OK.
If the seller pays the shipping company, the insurance contract is with the seller. You have no rights with the shipping company. The seller must handle the claim and you need to get your money back from seller.
Having seen it both ways, it made me want to be a good packer when I sell.
Edit: Overall the experience has been positive. Negotiate the deal up front (shipping, packaging, insurance, exactly what's included) and hope for the best. It's usually fine. Personally I don't think I'll buy another bass that's not in a hard case unless the shipper emails the pics of the packing process and I approve.