You choose two targets such that as a set, all of the targeting conditions are met. That means, for example, that if you control a nonland permanent whose mana value is lower than any of your opponents' nonland permanents, you can't choose it as a target. (If you did, you'd be unable to choose a second target.)

Even if you don't plan on exchanging control of any permanents, if you can choose two legal targets when the ability triggers, you must do so. If you can't, then you don't choose any targets.

If either one of the targets becomes illegal (because it leaves the battlefield or for any other reason) by the time the ability resolves, the exchange won't happen. If both targets become illegal, the ability doesn't resolve.