The controller of a card is usually the one who has it on their side of the field. The controller can change the Battle position of monsters they control if the monster is face-down then only the controller of the monster can see the card. Your opponent cannot see the face-down card unless if the monster is targeted to reveal the monster or they are the new controller. The effect of a card is applied normally if control is changed.

Example:

Player 1 Sets "Dharc the Dark Charmer".

Player 2 Summons "Destiny HERO - Defender".

Player 1 Flip Summons "Dharc", and uses his/her ability to take control of "Defender" and changes "Defender" into Defense Position.

Player 2 draws a card via "Defender's" effect.

Even though Player 2 owns "Destiny HERO - Defender", "Defender" is on Player 1's side of the field. Player 2 is Player 1's opponent, so Player 2 draws the card.

If "Defender" were to be removed from the field, it would be moved to Player 2's Deck, Graveyard, Hand, etc.

Cards such as "Reshef the Dark Being", "Creature Swap" and "Enemy Controller" are capable of switching the controller of monsters. Should said monsters be destroyed, they are sent to the owner's Graveyard. Some monsters, such as "Mataza the Zapper" and "Blindly Loyal Goblin" are immune to such effects.