This is then known as the Duelist's Deck Master. These Deck Masters are not classed as being on the field and stay face-up next to the player until the player decides to Special Summon it to his/her side of the field. The Deck Master's effect(s) are still active while not being played on the field.
If this were likened to be a real-world mechanic, it would be similar to the Side Deck; the only major differences are that the card is face-up and can be used and played during the Duel itself rather than in between duels. Other than these differences, the Deck Master is simply placed aside like the Side Deck.
The Deck Master can be Special Summoned at any time during the Duel. Each card has its own effect as a Deck Master.
If a player's Deck Master is destroyed, he/she loses the Duel. If a player's Deck Master is Tributed by a card effect to Special Summon another monster or used as a Fusion Material Monster or can otherwise be immediately replaced such as by the effect of "A Deal with Dark Ruler", the monster that is Special Summoned next becomes the player's new Deck Master.
If multiple Deck Masters are Tributed or used as Fusion Material Monsters, the Special Summoned monster is treated as the Deck Master for all of the players the original monsters belonged to.
All of the duels that used a Deck Master in the anime had at least one player summon their Deck Master to the field.