Originally Posted by LarryJ

There is no RADAR service of oceanic airspace. Airplanes are separated using non-RADAR procedures.

Non-RADAR separation is much the same as how trains are separated on the railways without the railroad controllers being able to "see" the train.

The flights are given specific routes and altitudes to fly and are required to make position reports at particular points. The position report will inlude the time and altitude at which you crossed the reporting point, your estimated time for reaching the next reporting point and the name of the reporting point after that. The air traffic controllers separate aircraft by not clearing them into the same areas, at the same altitude, at the same time. The required separation distance is much greater than when RADAR separation is used.

A lack of RADAR does not mean a lack of communication. Communication with ATC is maintained through satalite data link (new) or high-frequency radio contact through a 3rd party (usually ARINC, Inc.). Requests and clearances are relayed in this way.