Do you owe an insurance company money for the grace period?

Yes. If you add a vehicle within the 30-day grace period, you will be charged a premium for that time. In other Words, you still pay for the coverage even though you added the vehicle some time after it was acquired by you. The grace period just means that the insurer will still cover it between the time you acquired it and the time that you notified them (as long it was within the prescribed time limit) and not require you to notify them first. This would make it difficult to register newly acquired autos.