All the new aircraft deliveries I have been involved with were done by the buyer or the leasee. The normal process involves a physical inspection of the aircraft, paperwork, accessories, making sure the correct documentation is in hand, as well as one or more test flights to assure aircraft performance. In every delivery I have accepted, there is usually a squawk list, and the manufacturers are great at performing seeming miracles overnight. It is not uncommon that there is a couple of days of squawks. I often invite a factory test pilot along on the flights, as that helps identifying issues.
Some of the things I check include the rigging and trim of the aircraft, the power plant performance, the radio installations (which can require some coordination when picking up a plane mid-continent and testing HF), the autopilot systems, and the ECS.
Funds are transferred and paperwork is filed, and a flight to the destination is accomplished. Normally close fuel and power plant monitoring is done for that flight, and things like WX radar, strike finders and other avionics are extensively run.
If we don't have someone who has significant experience in the aircraft, we take training at a place like Flight Safety, in a simulator. Training is normally classified as initial or recurrent. The training outfit can tailor the training to include the specific equipment and engines that are included in the delivered aircraft.
The bulk of the aircraft I have personally accepted of delivery on are turboprops and jets, but it also includes a few piston aircraft. Some local flight schools, charter and FBOs have hired me to accept delivery of aircraft for them.
In my opinion the acceptance of new aircraft is more than just kick the tires and ferry the aircraft. It is a through systems checkout, and verification that all documentation is in place, cosmetics are good, and so, verifying that the end user receives a plane which is unlikely to have any issues, and that it performs as advertised.