I've sent out a few bottles of 7 beans from the Poké Pelago and received one bottle with 7 beans myself.
Is there any advantage to sending out bottles of beans or is it a gift mechanic for a randomly selected player?
Based on the page for Poké Pelago on Serebii, it looks like this is just to help random players over the Internet. It won't give you any benefit except the joy of knowing that you helped a stranger.
You can also, once a day, send out a Bean Bottle. This will put 7 of your standard beans out and distribute them across the Internet to other players and you will sometimes find Bean Bottles from other players turn up on your game.
On days when I send bean bottles, it seems like I have more pokemon trying to come to my island for me to "see if they will go into a ball". I thought it was supposed to boost luck on your island.
On days that I send one I will find at least three bottles washed up on the beach. They usually have rare beans in them.