Not true - the NHS negotiate bulk discounts as a large customer, but private medical care is available here.
When the government negotiates prices, it's called price-setting by economists. That's because it is a monopsony. I can't find any data on drug prices under private insurance in the UK, but it would be completely distorted anyway. Since the NHS remains a backstop for everybody, there is no reason for a private insurer to have a formulary that overlaps with NHS if their drug prices would be higher. It may be the case that UK private insurers offer drugs that NHS does not offer, but in that case, I would expect them to be in the same boat as US private insurers (who of course do negotiate drug prices, but have to compete with each other for customers, and those customers care about what and how much is covered on the plan's formulary).
Are private healthcare providers getting a free ride when they negotiate bulk discounts?
No, that's called a free market. The government in the US does not interfere in the market for prescription drugs directly. That's the point of not letting Medicare or Medicaid negotiate directly. It wouldn't be negotiation so much as dictation. The government only demands "most favored nation" status, meaning that it pays no more than the lowest cost offered to a private insurer.
Now, it's true that the US government interferes with the free market indirectly, but mostly in a way which increases demand for prescription drugs and other medical technologies, which boosts expected profits for health care companies, which boosts expected returns on health care investment, which then boosts health care investment. The US consumer and taxpayer pay a lot, but the rest of the world benefits from an implicit subsidy by seeing faster medical advances than they would otherwise.
All the NHS (and the healthcare providers in other rich countries are doing) is using the fact that they are large customers to negotiate a good deal.
No, they're using government monopsony power to free ride off the US consumer. The US government could put a stop to it, either by demanding "most favored nation" status for Medicare and Medicaid globally, or by allowing Medicare and Medicaid to negotiate down prices. Medical progress would slow however.
Here's a paper detailing some of the US inefficiencies:
http://www.ncbi.nlm.nih.gov/pubmed/15088673
Meh, seems like a small part of the problem. Regardless, most of the problem is caused by having 50 different regulatory agencies, one for each state. Also, because private insurers in the US don't insure people over the age of 65 (that's what Medicare is for), they end up insuring a group of people with lower average claims than the population as a whole. That makes administrative overhead look worse, since the denominator in the calculation is lower. Medicare's overhead is 2-3% I think, but the greater efficiency implied by the headline number is mostly an illusion.