If you place an insurance claim it will be the insurance company's claims adjuster that would decide fault and how much each driver is found responsible. If there are 2 different insurance companies involved then both would be determining fault and how much they calculate each motorist to be at fault or if one person is wholly responsible.
Fault should be placed on the driver that caused the accident if there is a clear cut motorist at fault. For example if another person ran a stop sign and hit you then normally they would be cited by the police for failure to stop and found at fault by the insurance companies.
Some states allow for comparative negligence which when applied in accident cases it is to determine responsibility and damages based on negligence of all parties directly involved in the accident. So depending upon the type of accident it could be possible that both you and the other driver could both be held responsible in different degrees.
For instance in Ohio they have comparative negligence laws that state parties share the cost of damages from an accident in proportion to the share of negligence. A motorist determined to be 50% or less at fault may recover their damages minus the percent cause by their own negligence. If more than 50% there would be no recovery for losses from the other party.
Other states, such as North Carolina, apply contributory negligence which means that if you are even slightly at fault in a collision you are not entitled to any recovery against the other driver.
Ultimately for claim purposes the insurance companies involved will investigate the accident claim to determine fault. This is done by the insurance adjusters whose job it is to determine the extent to which you are covered and the degree of fault of each driver. Sometimes an adjuster will split up fault differently than the police or determine fault to be placed on a different person then what the police report states.
An adjuster's investigation and determination of the claim usually takes into account both drivers' recollection of the accident, any witness statements and the damage of both vehicles. Insurance adjusters will also be guided by state laws.
Many times a claims adjuster's findings will agree with the accident report but there are times when it does not and they assign blame differently then the police. To find out about your state insurance and negligence laws you can contact your state's insurance regulatory body.