When travelling in state with a NFA firearm you are subject to the normal laws regarding firearms in your state. In South Carolina for instance, the firearm, if a handgun, cannot be concealed without a concealed weapons permit, and firearms may not be carried openly in public. These laws are subject to change by the legislature and vary from one state to the next.
When travelling interstate, that is when crossing a state border, with a NFA firearm you are subject to the laws of both states and have a duty to the ATF. Currently, the ATF requires anyone transporting a short-barreled rifle, short-barreled shotgun, machinegun or destructive device to file a Form 5320.20 (Form 20) with the ATF. Before travelling interstate you must receive the approved Form 20 and keep a copy with the firearm during your travels outside of the state in which the Trust is administered.
A Form 20 must be sent in duplicate to the ATF and requires the following information:
The name and address of the registered owner (i.e. the Trust)
A telephone number of a Trustee responsible for the transport
Notification if the travel is temporary or permanent
Dates of travel (The form can be for up to a year)
Information on the firearm to be transported
Reason for transporting
Where the firearm will be leaving
Where the firearm will be taken
How the firearm will be transported
The filing and subsequent receipt of this form will satisfy your obligation to the ATF, but you must still follow all state and local laws appertaining to the firearms.
It is worth explicitly noting that the ATF does not require a Form 20 for the transport of a silencer or ‘any other weapon,’ but you are still subject to state laws.