There is no comprehensive source of open data as you describe, but there are public and restricted-access open data sets that can illuminate certain aspects of private company finances. Privately-held companies often disclose information to the public in accordance with federal, state, and local laws. Some of this information directly involves company finances (such as public bankruptcy records; public property records) and some of it may serve as an indirect indicator of a company's financial circumstances (e.g., the U.S. Department of Labor publishes disclosure forms with rich information on the retirement plans maintained by U.S. employers, including 401(k) plan assets).
Certain sectors of private companies are subject to more comprehensive financial reporting, such as banks (see, e.g., the extensive federal open data on U.S. banks published by the banking regulators). Data.gov's search feature can allow you to explore data available on particular sectors, such as finance.
Depending on the purposes you have, you may also be able to take advantage of restricted-access data sets on individual firms. For example, the U.S. Census provides researchers with access to nonpublic data at a number of Census Research Data Centers (RDCs). Restricted use data includes economic data for business establishments and firms, including:
- Economic Census
- Longitudinal Business Database (LBD)
- Annual Survey of Manufactures (ASM)
The list of Census restricted use data sets is available on the Census Center for Economic Studies site.