I got some clarification on this a few weeks ago. The team with the worst record from the previous season is at the top of the waiver list. Once a claim is accepted, that team goes to the back of the waiver line.
The order of players ran through the list of teams is random. So each player goes through the wire and the highest team on the list at that time that put a claim in gets that player and then moves to the back of the line.
So in your case, this other owner in your league would get the first player that he claimed that goes through the wire.
I know this is kind of clear as mud, but hope this helps a little.