Here is a simple hypothetical example of how it could work.  Consider needing a test that medical protocols say should be done within 3 months.  The public system continues to function within that protocol (i.e. standards can't be allowed to fall), but as a patient I want to do to the test today and incur a cost to do it.  This gives me what I want, it frees up resources on the public program, and a patient on the public system now has one less person on the list in front of them, thereby reducing the time they need to wait.  If done well, this is a win win for all.

This is highly complex, but it could be done.