It is illegal to sell/transfer a gun, except in very limited circumstances, without going through a dealer for the sale/transfer (background check and waiting period).
A private sale is a sale between two person, who are not licensed firearms dealers (FFLs). Many years ago, it was legal for two people to sell a gun without any paperwork or the involvement of a dealer. In a misguided attempt to stop gang members and drug dealers from buying guns from street dealers, California criminalized private sales. Background checks have a high false positive rate for denied checks.
 
There is no way to purchase a firearm “off the books” in California. There is no such thing as “the gun show loophole” or “Internet loophole.” Unless someone wants to commit a crime, you cannot obtain a “paperless” gun in the state. You cannot have a firearm you buy from a private party off the Internet delivered to you except through a dealer. Buying a gun privately online means going to an online gun forum or classified type site (think Craigslist, but for guns) and arranging to purchase a gun from someone. This is in contrast to going to the gun store and picking a gun off the wall or from under the glass.