Actually I think it would be the seller's responsibility. If you owned a gun in this state and sold it, wouldn't you want to be sure that this sale was reported? If you buy from an FFL, they are the one who submits it to the CHSB, not you.
Interestingly enough, the form does not say who must submit the original copy.
Should a special form be used when selling guns through a private sale?
A: The state form required for private sales is an FA10 Form. The FA10 replaced what was known as the "Blue Card" previously used for private sales. This is essentially the same form used by dealers except that instructions are included with each private sale form. This form can be obtained from any police department. The seller must fill out the form and forward it to the Firearms Record Bureau (FRB) within seven days of the transaction.