There isn't really any way to make that comparison because Medicare provides a different type of benefits to a different subset of the population. Private insurers don't offer policies to the elderly that cover the things Medicare covers because those things are already covered by Medicare.

People sometimes try to measure things like "overhead" but that also doesn't really work. If one entity pays $10M in claims but half of them are unknowingly fraudulent and another spends $1M to prevent the fraud, the first one is paying $10M with 0% overhead for fraud detection while the other is paying $5M with 20% overhead for fraud detection. Comparing the percentages implies the second one has "higher overhead" but it's also the one paying $6M instead of $10M.

And the same thing bleeds into the prices for specific services. A provider will happily give you a "20% discount" on the price if you don't bother to check that half their claims are fake. Then you're actually paying them 60% more while the books show them charging you 20% less per procedure.

On top of that, the market for many medical services in the US is extremely captured. The majority of states still have Certificate of Need laws, which are literally laws prohibiting new providers from entering the market if the local government (i.e. the incumbents) think the result would be "too much" competition. The AMA consistently lobbies to limit the number of medical residency slots and sustain a doctor shortage, and doctors who immigrate from other countries, even licensed physicians in countries with first rate healthcare and medical education systems, can't practice in the US without doing a US residency, which in turn consumes one of the residency slots and prevents immigration from alleviating the shortage whatsoever.

The premise is that competitive markets result in lower prices, not that uncompetitive markets do.