When a player is on a winning streak in blackjack, etiquette rules suggest maintaining a sense of humility and sportsmanship. It's important to avoid gloating or celebrating excessively, as this can disrupt the atmosphere at the table and make other players uncomfortable. Players should also continue to respect the dealer and fellow players by following game rules and avoiding distractions. Sharing a portion of winnings with the dealer as a tip is a common practice that acknowledges the dealer's role in the game.