Okay I just skimmed this so forgive me if anything is redundant. I am a former oceanic controller at ZNY. There are 3 ARTCCs with true oceanic. New York, Oakland, and Anchorage. Those other centers mentioned have some non-Radar over water, but they are not oceanic as defined by separate Flight Information Regions (FIRs), and they do not use the system I will describe below. The controllers in the United States sit on the same control room floor as the domestic controllers, and once reaching full performance level also certify on radar sectors as well (I only certified on the ocean sectors). In Gander, they are co-located as well but the oceanic and radar controllers are separate.
At ZNY, ZOA, and ZAN the oceanic sectors are worked on a computer called ATOP (Advanced TransOceanic Procedures). The computer integrates with ARINC, which in the case of ZNY is in a building across the street. In New York the sectors are divided geographically, and in New York, there are two different areas responsible for WATRS airspace and the NATS airspace. I worked the NATS side. The WATRS side could be split into as many as 4 sectors, but typically on an average day it was split down the middle with one person working the east half and one person working the west half. Bermuda Radar and east oceanic (out to 40w) as well as the off-shore radar sectors in the vicinity of W386/105/107, were the responsibility of the area I worked in. The non-radar airspace I worked was divided into 9 sectors that could further be divided east/west if necessary. Typically on an average day that entire airspace would be split 2 or 3 ways, however on a busy track day or night, I have seen it split into as many as six separate sectors, and the area had enough work stations to split at least 8 ways (if I remember correctly).
The oceanic computer might look similar to a radar to a layperson, but it is not even close. The airspace GUI was for situational awareness only. Everything is handled via position reports that are either uploaded to the computer automatically via ADS, or manually by ARINC. When I left, there was talk of somehow implementing ADS-B over the ocean, but at the time I left it was ADS-C, I have not kept up with the advancing technology since I left. the -C stands for "contract" and what that meant was that the pilot had to log on with the appropriate facility, and that facility would then tell the onboard computer how often to make reports. Clearances were issued via CPDLC or ARINC relay. If necessary, at the request of the controller or flight crew, direct voice communication could be established via ARINC HF, but that is only done in unusual situations.
The computer takes the position reports and checks for potential conflicts. If aircraft are anticipated to lose minimal applicable separation within 2 hours, it flashes orange. By the SOP the controller may take action at that time, or they can wait it out. Once the computer predicts separation will be lost within 30 minutes, the involved aircraft will flash red and the controller is required to take action. The computer tells the controller when separation will be lost, but does not make any suggestions on how to fix the conflict. It also requires human input to recognize and account for certain situations where reduced separation minima can be applied. Required separation varies based on the type of aircraft and the equipment onboard. Turboprops require twice the separation as jets, typically. When I left, standard separation for jets was 50NM laterally and 10 minutes longitudinally at the same altitude; however that could be reduced to as little as 30NM in any direction, and I understand it was reduced further after I left. To qualify for 30 NM separation, both aircraft had to be equipped with RNP4, ADS-C, and CPDLC.
Hope this helps
@Kingairer