It does not matter who threw the tile. A winning hand in Mahjong Island is 14 tiles that match one line on the Island Card. The Island Card is Mahjong Island's own card of hands.
What does winning on a discard look like?
You take the discarded tile, and your hand is complete. Say your line on the card needs a Set, three matching tiles, of 7 Dots. You hold two 7 Dots. Another player discards a 7 Dot.
If that Set was the last thing your hand needed, you take the tile and win. Taking another player's discard is called Take Tile.
What if two players want the same tile?
A Win comes first. Say one player wants the tile to win, and another wants it for a Group of matching tiles. The player who wins gets it. Otherwise, the player closest after the discarder gets the tile.
You can also take a discard to make an Open Group. See Do You Draw After Taking a Tile?
Is drawing your own winning tile different?
Yes, a little. You can also win with a tile you draw yourself from the Draw Pile. That is a Self-Drawn Win, a bonus worth +2 Mult, a score multiplier.
Watch the Self-Drawn Win lesson, or see every rule in Mahjong Island Rules: How to Win a Hand.