Usually with NTS the originating station, which is always a ham radio station, receives the acknowledgement of delivery which is usually returned via NTS message using ARL numbered phrases like ARL 47 means delivery was successful and ARL 67 is not delivered for a stated reason. An example of non-delivery reason would be bad phone number as many times a phone number of recipient is included in the message delivery instructions.
NTS has a network of layered message handling sections that include local nets, section nets, and regional nets. A regional net might be something like RN6 or RN7 and a section net is something like OSN (Oregon Section Net) and WSN Washington State Net. I only mention OSN and WSN because I am a member of those nets but there are others in North America.
If for some reason you can not deliver back thru the NTS messaging system then another form of acknowledgement can be done and you are open to take your pick of phone call, e-mail, or routing back to who you actually received the message. This last method is often the most common but not the most formal which is to use the NTS messaging system.
In answering this question, I am answering from my own experience in NTS message traffic handling and not because I just read it in the ARRL documentation. I say this because it has been years since I read the ARRL NTS documentation and there could be more detail and even changes from the methods I and others I know employ.