Sorry can't say for sure, but you can actually easily test it out.
- Quicksave, and take two planets and jot down their research output.
- Rushbuy that 'Training Center' on one of those planets.
- Now compare if the research has been upped a] only on that planet, or b] on the other as well.
If b] is the case then that bonus is civ-wide, and will also apply to all your other planets as well.
If your findings doesn't suit you simply reload the quicksave 