…But when you submit an entry on paper to a contest that says “All entries become the property of Contest Enterprises and will not be returned,” that does not mean that the contest operators have intellectual rights to your composition. It just means that they own the paper you submitted. This gives them the right to throw it away.
In all likelihood you have not given away any of your rights as author unless (as Dominic says) you signed or agreed to something that surrenders your rights. You should be able to submit the same work to any number of contests. Note, though, that the status of your entry may change if it is a winner. especially if the prize includes publication.