Many doctors have decried private-equity firms’ push into healthcare, saying patient care has eroded under their ownership. Now, as these firms make similar inroads in accounting, some medical professionals have a warning for CPA firms: Don’t make our same mistakes.

Drs. Bob McNamara and James Keaney, both former longtime emergency-room physicians, were early critics of private equity in healthcare who now say such ownership in accounting could shake trust in an industry that investors rely on for getting trustworthy financial information about companies.