As background information, all of our client's members are organizations. I see that if I select "on behalf of organization" and "membership section enabled" I get two copies of the profile on my page. So we won't do that.
However, neither option by itself records a membership in the membership dashboard when the profile is filled out in our tests. I have all the key fields; organization name, email, contact address and phone. I get no errors when saving the contribution page. I have no errors when submitting a live form. What am I missing?