The item basically makes you do 2 autoattacks at the same time. If you have an item that procs on every auto attack (not ability hit, auto attack hit), then it will proc that item for that bonus attack too.
Take Tiamat and Sheen for example;
Tiamat works on every attack. When you proc the double attack, Tiamat will hit both times.
Sheen procs for one attack and goes on CD. Your first attack will use up the proc, and the bonus attack won't have it to be used so will just be a plain auto attack.
I'm gonna have to look for Rioter post to confirm this though lol. I do recall they mentioned him somewhere.