Users who played the Modern Warfare Beta on PS4, who have linked their PSN, Xbox Live, and Battle Net accounts on the same COD Account, are reporting that their levels and progression are showing up on all platforms.
The game is confirmed to support Cross Play between all platforms, but Activision and Infinity Ward said they were just "exploring" the idea of Cross Progression saves. They have not officially confirmed it as of now.
I hope this sticks around for the full release but this is a pretty great development.