Doesn’t an immunity deal mean you agreed to testify in return for not being charged?
Yeah, you don't get an offer of immunity unless you have "proffered" something in return. Typically it's pretty big, as in you promise to tell everything you know, down to the last detail, and you promise to be absolutely truthful, hold nothing back, and commit no lies of omission.
And if they catch you lying or holding out, they can go to the judge and get the whole immunity deal thrown out and put you on trial for the original charge, or a more severe charge.
It's all part of that famous prosecutor game: The First To Squeal Gets The Deal.
“Doesn’t an immunity deal mean you agreed to testify in return for not being charged?”
Yes, but there is limited immunity and full immunity. Haven’t heard which one Pagliano received.
Pagliano was given limited immunity. See the aticle at the link. He could still be charged.