In my experience buying auto insurance, 25+ years now, I don't remember negotiating premiums with agencies. They gathered my info, gave me a quote with my desired coverages, showed me some additional discounts I may qualify for by adding other products, and that was it. I don't recall going to insurance company B and showing them the quote from insurance company A, then having B price match or beat A.
What I do remember is company B trying to beat the quote from company A by offering less coverage and/or higher deductibles. So while it looked on paper like I was saving money, it wasn't an equal comparison.