State privacy laws and insurance guidelines differ however a competing insurance carrier is not normally allowed to have access to your current insurance policy. No one has access to your specific policy. Typically, for marketing a company will look at your current registered vehicles (public information) and try to rate it with that. Competing insurance companies/agents can access general information about you like where you live, your age, tax status, etc., but they should not have access to your actual policy without your permission. If they do, then they are in violation of insurance privacy regulations.

Most likely the insurance company that called your husband was using a marketing scheme to get your business. As for comparing their rates with your insurance company's rates overall, that is possible since some companies do publish sample rate quotes. For example, it appears you live in Kentucky, here the KY Department of Insurance (DOI) offers a consumer guide that lists companies authorized to sell auto/homeowners/renters insurance in Kentucky and those companies have provided rate comparisons for certain situations. If you are interested in Attached viewing those rates you can go online with the KY DOI.

If you and your husband believe that the insurance carrier that contacted your husband truly did have a copy of your policy then you can have the Kentucky DOI, the insurance regulator for the state, investigate the issue and you can file a formal complaint with this state agency. 

If you are shopping around for insurance from an auto insurance specialist, you can follow this link for rate quotes.