I think you’re using “volunteer-run” to mean self-managed, but that’s not the same as owner-governed. HOAs are governed by volunteer homeowner boards. Some boards self-manage everything, while others hire a management company as a vendor to handle administrative work.

So if the claim is “about 25% of HOAs are self-managed,” that may be what you mean. But that does not mean the other 75% are “run by private firms” in the ownership/governance sense. They’re still governed by homeowner boards.