It can be difficult to get insurance if someone else finances a car for you for a couple of reasons. One is that you do not have insurable interest. Since the other party is the one with the car in their name (and thus the registered owner) than you do not have insurable interest in the vehicle. There are insurance companies however that will allow you to obtain insurance though you do not have insurable interest in a car if you have care, custody and control of the vehicle. Esurance is one such company.
The other issue you will run into is that the finance company will likely require the person who financed the vehicle to be the one that holds the insurance policy on the car and thus making it difficult for you to get the insurance. The financing company (lien holder) has say over the insurance requirements since the car is their asset and so can require the party that financed the car to hold insurance on it.
So when you are driving a car, that someone else financed for you, and want to insure the vehicle yourself the problem that arises is that the finance company usually wants to have evidence of insurance in the name of the person responsible for the loan and not in your name. If you are the named insured but not the name on the loan than the lien holder may require that the person on the loan also is on the insurance policy as a named insured.
One way to get insurance that covers you for this car that is financed under the other person's name is to have that person get the car insurance and list you as the principle operator. If their insurance provider understands the situation as you being the primary driver for the car your friend (relative, etc) financed than you should be covered properly and you could pay the owner for the insurance policy on the vehicle.
Since insurance companies such as Esurance will insure you without insurable interest than the person financing the car would need to read through their loan documents to see how the insurance is required to be carried and even speak to the lien holder to see what they will accept as proof of insurance.
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