Yes, but if you look at the map, the state is yellow, which is "Legal" for private transfers. I do not have cites at present, but from what I remember, private transfers are legal, with limitation of no handgun transfers to those under 18. It is unlawful to knowingly
transfer firearms to prohibited persons. Private parties May
use the BG check system to determine eligibility, but there is no requirement
to do so.
Clark county is its own for the registration. From what I recall, to sell to another Clark County resident, you must make the transfer at an LE location so the registration is recorded.
Disclaimer: IANAL, nor do I play one on TV, and I did NOT sleep at a Holiday Inn Express last night.