As respects the liability exposures, you should be included as an additional insured under the owners insurance policy. This would be done by an endorsement issued by the company. As respects damage to the aircraft you are operating, you could enter into an agreement with the aircraft owner wherein the owner agrees not to hold you responsible for damage you cause to the aircraft. The owner would then be required by his insurance policy to advise the insurance company of the agreement, obtain their approval, and have the insurance company issue an endorsement waiving their rights of recovery against you by way of subrogation for damage to the aircraft resulting from your negligence. For confirmation that the company has agreed to protect you, you need for the company to issue a Certificate of Insurance to you showing you as being included as an additional insured, the waiver of subrogation, and 30 days prior written notice to you if the company cancels the policy.
Different companies take different approaches to such a request.
It depends often on how they are approached. This information is directed to persons who are providing professional pilot services - freelance or contract pilots. Pilots who are regular employees of the named insured are usually included as an insured by virtue of their employment status. I hope this information is helpful. Remember, I am not an attorney and the information above is general in nature.