All players get to see which card you chose at random.

Except for the card that you return to your hand, reordering your graveyard won’t affect anything that’s currently targeting or enchanting a card in your graveyard. Whatever it is will be able to track the card it’s targeting or enchanting as your graveyard is reordered.

The easiest way to choose a card at random from your graveyard is to turn your graveyard face down, shuffle it, and pick a card. You’re not normally allowed to reorder your graveyard, but since this card lets you do it anyway, it’s okay to mix the cards up while you’re picking randomly.

If you have multiple cards in your graveyard with the same name, and one of them is being targeted by another spell on the stack or enchanted (with Spellweaver Volute, for example), you must differentiate them so you know which one (if any) is chosen at random. In that case, it may be better to use dice to choose a card at random, or temporarily use a proxy card to represent the targeted card.