Adtran debug is different from Cisco in this regard. The output streams only to the session that entered the debug command.
My practice is to open two SSH sessions. On one of them I'll do a "no events" and use that as a clean window for command and observation. The second one I'll use for the debug output. If it gets so noisy that it's overrun I can just kill it. This also turns off the debug process attached to the second session. Unlike Cisco it doesn't stay running in the background which is a good thing IMHO.