It was Barzini who set up the meeting with Tattaglia and the other families.

"I didn't know until this very day that it was Barzini all along"

It was the same scenario with Barzini setting the meeting with Tessio.

In that world you're suspicious of everyone, including those that are close to you. More so probably because they can hurt you the most because they are close. If any of your top guys are meeting with the enemy you would be blind and stupid not to be aware of what could happen. So his antenna is already up and looking for trouble. With Barzini's past it's pretty easy to put 2 and 2 together if you're paying attention.

As Yogi Bera once said "You can observe a lot, by just watching"