There are many ways to gain control of a permanent that is under another player’s control. For example, Traitorous Blood will do so temporarily, while Mind Control will do so as long as the Mind Control remains attached to this creature.

It is important to note that you do not gain control of any equipment or auras that are attached to a creature when you gain control of it. In some cases, this doesn’t matter; however in others, it can completely change how things work.

With regards to equipment, since control of it does not change, the player who does control it can still use its equip ability to move it to another creature (but still only on that player’s main phase). You will not be able to use the equip ability.

Some abilities of equipment and auras will work differently–or not at all–when the creature they are on changes controllers. For example, if I control a creature enchanted with Curiosity and you steal it, the Curiosity will not do anything if this creature damages me. This is because Curiosity’s ability triggers when the enchanted creature deals damage to an opponent. I still control the Curiosity and cannot be my own opponent, so this ability will not trigger.

It gets even weirder with the Swords. If you steal my creature equipped with my Sword of Body and Mind and attack me with it, I mill the top 10 cards of my library, but I also get a Wolf token. The distinction here is between the wording of the two abilities, where one says “that player” and the other says “you” (remember “you” always refers to the object’s controller).