The exchange occurs only if Conjured Currency and the target permanent are both on the battlefield and controlled by different players when the ability resolves. In addition, the target permanent must be one you neither own nor control. If any of these things aren’t true when the ability tries to resolve, the exchange won’t happen and neither permanent will change controllers.

The only legal target for Conjured Currency’s ability is a permanent both owned and controlled by another player, although it doesn’t have to be the same player. For example, a permanent owned by an opponent but controlled by your teammate (or another opponent) could be chosen as the target.

You don’t necessarily have to control Conjured Currency when its ability resolves in order for the exchange to happen. If another player has gained control of Conjured Currency while the ability is on the stack, and a third player controls the target (and you don’t own it), you may choose to have those two players exchange control of the relevant permanents.