Illegal self duplication is easier to control than enemy duplication. If I dispel a bear strength on your Mage, and three rounds later you reveal another bear strength: either you have not cast any enchantments on yourself over the past three rounds, in which case I call you out for illegal casting, or you have cast enchantments on yourself in the meantime. In that case, you've spent a quick action and 2 mana so you've effectively paid the cost of re-enchanting.
That said, in an extremely high stakes tournament game, if one player has a large number of face downs, it could be reasonable to call a referee to make sure there are no duplicates.