Just like they read, they kick in when they say they do, like during your opponent's turn for Bidoof's Wacan Berry or at the end of your turn for Pikachu's Oran Berry or Shinx's Rawst Berry. Note that the key phrase is "at the end of your turn" for stuff like Rawst Berry; since it cures burns at the end of your turn, you don't flip for the burn check that happens in the between turns stage after your turn ends.
Likewise, Pikachu's Oran Berry only heals it 10 at the end of your turn, not between turns.