If you wrote the poem.. effectively you own both the poem and the copyright to the work. Where you publish your work is up to you, but do not (if you are serious about your poetry) sign away your rights to your work. Always read the TOS of every site that you contribute to, to ensure that you do not give away your rights unwittingly.
Now, in regard to your concern here..
You may well be pinged for duplicate content at HP and within the Search Engines and you may not. In such cases the site at where the works were first published is deemed to be the original publishing party and as such can (though for only occasional breaches often don't) endorse a points penalty and alike for such. Entering in poetry comps, is used a lot as a site drawcard, merely to gain site content and for the unsuspecting... they must consider which site best supports their objectives.
So.. Good Luck with your poem... but do keep in mind that Duplicate Copy is Not Original Copy and as such carries with it a variety of penalties. Now if you change the poem to 'not be seen' as such....
You choose - you do whatever is best for YOU - Ethically and professionally and don't second-guess your decisions!
Write up a storm and hold it within the words of your poems.