Bounty does two things. First, the pirates attack whoever has the highest bounty. The amount of bounty and a few other misc things (number of that player's colonies/trade ships) determine how big the pirate raid will be. They start off small, but grow bigger as the game goes on.

The second is that a player can collect on the bounty of another player. When you put bounty, it gets split between everything that player has. Not evenly, of course, so taking out a planet will award a bigger chunk of the bounty than a frigate, for example. Note that this doesn't do anything in 1v1 because you can't collect the bounty that you yourself put.

So in your case against only one opponent, the only thing bounty will do is make the pirates attack your enemy. They will always attack whoever has the highest bounty at the time the raid launches. You get a notice that it's going to launch soon once the timer reaches the end, and then there's a 60-90 second window (I think, memory's a little rusty) until the raid actually launches. If the other guy has a higher bounty than you when the raid launches, it goes after him