The first broadly useful fully open source models will do this.
We already have open data / open code / open weights for some domain-specific cases, such as audio models trained on large open datasets, eg. Tacotron / LJSpeech from waaay back in the day, though that is certainly not SOTA anymore.
Distillation could possibly be considered an early case of this as raw AI outputs are themselves not copyrightable unless humans enrich, filter, or transform them. Granted, that does not handle the cases where the outputs are sufficiently similar to copyrighted original works.