There is no doubt that Medicare has the bargaining power to drive down hospital costs compared to private insurance: Medicare gets 10c/$1 on the chargemaster at hospitals (and hospitals set prices so that Medicare's payment rate has a small profit), while private insurance tends to be able to negotiate 15-20c/$1. Individuals who are good negotiators can reportedly end up paying only 20-30c/$1, so if you have a big hospital bill, it can pay to call them up and negotiate.

Retirees on the private market get completely shafted, though: almost everybody in their age group uses medicare, so the insurance will need very high premiums to cover their risks. From what I have heard, people who are on employer plans at that age have better health outcomes than retirees, regardless of where the retiree gets their healthcare.