Who decides whether a throw-in was taken from the right place?

The referee, not you. Under World Rugby Laws 2026 (Law 6.26d), it is for the referee, and not the assistant referee or touch judge, to decide whether the ball was thrown in from the correct place.

Your job is to stand at the place of the throw and point to the team entitled to throw in, and to lower the flag when the ball is thrown in (apart from the exceptions listed in the touch signalling question).

Source: World Rugby Laws of the Game 2026, Law 6: https://passport.world.rugby/media/wnvjlxtb/2607en-world-rugby-laws-final-compressed.pdf