Can you transfer a self-built firearm (that has no serial number)?
Per the ATF, you can make your own firearm for personal use, but cannot sell or distribute it...regardless of whether you put a serial on it, or not. A serial number would really only be useful in case it's stolen, which can show that it was yours and the person who stole it didn't build it themselves.
"For your information, per provisions of the Gun Control Act (GCA) of 1968, 18 U.S.C. Chapter 44, an unlicensed individual may make a “firearm” as defined in the GCA for his own personal use,
but not for sale or distribution." (Emphasis from the ATF's website...they actually bold this on their site)
http://www.atf.gov/firearms/faq/firearms-technology.html#commercial-parts-assembly
Confusingly, later in the answer, the ATF says this:
"Individuals manufacturing sporting-type firearms for their own use need not hold Federal Firearms Licenses (FFLs). However, we suggest that the manufacturer at least identify the firearm with a serial number as a safeguard in the event that the firearm is lost or stolen. Also, the firearm should be identified as required in 27 CFR 478.92
if it is sold or otherwise lawfully transferred in the future." (Emphasis added by me)
So...you can make it for your own use, but not for sale or distribution...but you should identify it as required if it is sold or lawfully transferred. Weird...