As a biller and small business owner, the contract is the document that outlines responsibilities. Though, the short answer is that billers depend on providers and/or their office staff to provide correct information regarding patient demographics and insurance coverage.
The longer answer:
Does the office staff verify a patient's benefits, or is that something the billing company is contracted to do? If the office staff verifies benefits, then, naturally, they would be the ones who would notice an error with the insurance and would then be responsible for getting the correct information to the biller. If the billing company is verifying benefits, then they would notice an error because, without the correct information, the patient could not be found via a portal, or by the insurance company's customer service rep. When this happens to me, I inform the provider/staff and request that they obtain the correct info from the patient and then forward it to me, so the benefits can be verified. If no one is checking benefits, this should be remedied immediately.
In general, the provider's office is responsible for sending the billing company the correct demographic information. Since billers usually do not have direct contact with patients, office staff is best poised to obtain and update this information, and notify the biller of any changes. The billing company would be responsible for updating its own records to reflect accurate and current information. Is the billing company logging into the provider's software to submit claims? (I do not recommend this at all, because it is a recipe for disaster.) In this case, it should be outlined who is responsible for updating or correcting patient demographics and insurance info in the system.
Ultimately, though, good communication is key for a smooth billing process -- regardless of whether the billing is done in-house or by a third-party.
Hope this helps!