Both people you talked to are describing different sides of the same statute. In most states, managing real property for another person for compensation is a licensed activity, requiring a real estate broker license or a separate property manager license depending on the state. Many of those same states carve out an exemption for a salaried on-site employee who works for the owner of that one property, which is why the site manager you met isn't licensed. A management company taking over a 120-unit under a contract with an owner is usually on the licensed side of that line, and the entity itself often needs a license with a designated broker or qualifying broker named. A few states regulate this lightly or not at all. Which category your arrangement falls in depends on how your state's statute reads and how you structure the contract, so confirm it with your state real estate commission and a local attorney before you sign anything.
On insurance, you carry your own. General liability and errors and omissions for the management company, workers compensation once you employ site staff, and a crime or fidelity bond because you'll be handling other people's rent and deposits. You also get named as an additional insured on the owner's property and liability policies, which protects you on claims arising from the building itself.
@beacon CPM and CAM are professional designations, not licenses. They don't authorize you to do anything, and institutional owners' management RFPs ask for them anyway, so they affect what work you can win.