They don't reveal who pulled the trigger in the movie.

I think Haydu did do it though. His expression in reaction to Chris saying the name Dickie Moltisanti was one of "....oh shit." He's also screaming "I'm Sorry! I'm Sorry!"as he's crawling away right before Chris shoots him. I don't think Tony would have Chris kill a cop, retired or otherwise, just to tie them closer together. Tony could have just as easily ordered Chris or anyone else to kill Barry and not given a reason to do it. We don't hear anything about Barry before or after, he's a retired cop. Could have possibly been on the payroll. Tony alludes to the fact that he was "useful". In the podcast "Talking Sopranos" they ask Chase about this and he said that he wrote the scene as if Barry was definitely involved, but also that Tony may have not been 100% honest with Chris about the circumstances. Obviously Tony wasn't involved when Dickie was killed as he was a teenager and we can surmise that Tony probably wasn't aware of who was behind the killing or why. Even if he got confirmation Barry pulled the trigger.