All firearm purchases in the United States must be, in some form, registered with the underlying law enforcement agency of the particular jurisdiction in which the gun was purchased.
Gun laws in the United States widely fluctuate between states, so understanding your area’s particular gun laws will save you an enormous amount of legal hassle or potential trouble. That being said, the individual who purchases the gun is responsible for registering the firearm.
Depending on state laws, the process and requirements needed to legally purchase a gun will fluctuate based on individual state interpretation. Some states hold a very scrupulous stance towards the legal obtainment for a firearm, while others are more lax in the required obtainment process. Laws will also fluctuate based on the type of firearm purchased.
In general, registering guns for hunting firearms is more lax and in some cases, not necessary, whereas registering guns for handguns or semi-automatic weapons will undoubtedly require a more stringent process.