Private hospitals should release the same levels of data about patient safety incidents as NHS providers, a report says.
The organisation behind the report says it is difficult for the public to judge the safety of private hospitals.
Private hospitals now receive 28% of their income from treating NHS patients.
Independent hospitals say they are working hard to provide better data.
Private hospitals are required to report all serious incidents to regulators at the CQC and Monitor, and additionally to commissioners if NHS-funded patients are involved.
But it is not possible to compare data side-by-side from NHS and private hospitals for death rates or complications such as infections.
A think tank scrutinising the role of markets and competition in the NHS, the Centre for Health and the Public Interest (CHPI), says figures obtained from the Care Quality Commission (CQC) show just over 800 patients died unexpectedly in private hospitals in England between October 2010 and April 2014 - and there were more than 900 serious injuries.
The report says: "It is not possible to state whether these rates of death and serious injury are significant, as we do not know in which hospitals they occurred, the health status of the patients concerned, nor the types of treatments that were being provided.
"It is also not possible to state whether the rates are unusually high or to be expected."
Private hospitals' representatives say those figures need to be examined against comparable NHS data.
The report also says there were 2,622 emergency admissions to the NHS from the private sector in 2012-13.
CHPI is calling on the government to review the nature and costs of these admissions.
Report co-author, Prof Colin Leys, said: "The public and regulators have access to more information than ever before about how NHS services are performing but the same cannot be said for private hospitals.
"The government has recognised the crucial role of transparency in making hospitals safer, but reporting requirements should apply wherever patients are treated. This is too important to be left to the industry to address.
"This is a question of accountability. It's a basic necessity for all hospitals to report the same data."
The report recommends that private hospitals report on their performance in the same way as NHS providers - with data such as infection rates made available on the NHS Choices website, and monitored by the Health and Social Care Information Centre.
Peter Walsh, chief executive of Action Against Medical Accidents, said: "This report confirms it is time for the same level of scrutiny, regulation and protection of patients' safety and rights to be afforded to private patients as is now being done for NHS patients.
"A comparable complaints procedure and access to independent advice on complaints would be a good starting point."
Fiona Booth, chief executive of the Association of Independent Healthcare Organisations (AIHO), said: "From the data already publicly available from the CQC, it is clear that the care provided by independent hospitals is overwhelmingly safe.
"The independent sector has been working with the NHS for a number of years on a project to share and report quality data.
"A lot of work has been undertaken to make reporting systems compatible. That work is ongoing, and will enable the independent sector and the NHS to report incidents in a directly comparable way."
The Private Healthcare Information Network (Phin), which collates data, said: "Our network was set up precisely to improve the availability of information for patients, and has strong support from private hospitals, insurers, and doctors to do so.
"Most of the specific issues raised in the report are already being addressed, and indeed all private hospitals and consultants in private practice will soon be legally required to work with us to publish far more comprehensive information in the near future."