The Dayton Daily News reports on a lawsuit seeking the records of the Ohio Municipal League, on the grounds that it is a publicly funded body.
“The rule in Ohio is that if you are the functional equivalent of a public agency, you can be compelled to produce records, even if you are not, per se, public,” said Maurice Thompson, executive director of 1851 Center for Constitutional Law, a public interest law firm in Columbus that advocates Constitutional rights and limited government.
Laws differ state by state on this account. Here's a Michigan case, Eric Jackson v Eastern Michigan University, 215 Mich App 240; 544 NW2d 737 (1996).
Eastern Michigan University Foundation is primarily funded by Eastern Michigan University and, therefore, is a public body subject to FOIA.
The Michigan courts ruled differently on the case Breighner v. MHSAA, 662 NW 2d 413 - Mich: Court of Appeals 2003
Michigan High School Athletic Association (MHSAA) was not a "public body" under MCLS § 15.232 (d)(iv) because its current form was not created by a state or local authority and receipts from ticket sales for its tournaments did not constitute funding by or through a state or local authority; hence, the MHSAA was not subject to the Freedom of Information Act.
The biggest question that will repeatedly come up in these cases is whether the body in question is an agency, doing the work of another agency under contract, or whether it's simply an association with agencies as members.
The full Ohio law is here: ORC 149.43 Availability of public records for inspection and copying.
The Ohio attorney general writes:
The rights and duties set out in the act apply only to a “public office or person responsible for public records,” which includes governmental subdivisions, private entities that are the “substantial equivalent” of public institutions, and other “persons responsible for public records.” The act does not apply to private corporations or other organizations and is also different from the federal Freedom of Information Act (FOIA), which applies only to federal agencies.
I'm not as familiar with the cases in Ohio, but it appears that State ex rel. Oriana House, Inc. v. Montgomery, 110 Ohio St. 3d 456 - Ohio: Supreme Court 2006, and State ex rel. Repository v. Nova Behavioral Health, Inc., 2006 Ohio 6713 - Ohio: Supreme Court 2006, as relevant to the issues at hand.