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If your guests post photos from the wedding on Instagram or Facebook, they can add your #hashtag to the description. When you search for that hashtag on either social media platform, you should see all of the photos that your guests posted (as long as they listed the hashtag in the description!).
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Yes - if it's common enough, other people might use the same one. Otherwise, if it's super unique, there's less of a chance that someone else will be using it too. Unfortunately, there's no good way to guarantee that no one else will be using your hashtag, other than try to be as clever as possible. Something like #SmithWedding will likely have more people using it than something like #SmithenWithYou (as a play on "smitten with you"). You could always search for any hashtag ideas that you come up with to see if anyone else is using them.