As Augusta's last ability resolves, the player whose turn it is chooses a card to exile from their graveyard, then each other player in turn order does the same, then all of the chosen cards are exiled at the same time. Each player will know the choices made by the players who chose before them.

You don't choose a target for Augusta's last ability at the time it triggers. Instead, a second "reflexive" ability triggers whenever one or more nonland cards are exiled this way. You choose a target for that ability as it goes on the stack. Each player may respond to this triggered ability as normal.