Activision and Treyarch confirmed that the private beta will officially become available for PS4 players starting on September 10th, 2018. Timing for the other platforms and the open beta have not yet been announced.
See the Tweet announcement below.
? Call of Duty (@CallofDuty) August 14, 2018
Call of Duty: Black Ops 4 hits the PS4, Xbox One and PC on October 12th, 2018.