Other than the player who played the Tournament card, other players who have a Province in their hand are not able to choose a prize.
Yes, that's how the card is supposed to work, only the Torunament player can gain a prize! And that depends alone on the fact if they can reveal and discard a Province.
All the other players can do by revealing a Province is to prevent the Tournament player to get +$1 and +1 Card at the end of tournaments effect.
There is an option "Don't reveal", but no option to "reveal".
If you have a Province in hand in that situation, it will be marked with a green outline, indicating you can click on it to reveal it.