I'm generally in favor of it. Whether to include your manager or not is up to the team dynamic. On our team, absolutely (she used to be a dev on the team, then the team lead, then the team manager).

What I've seen in our own "private" room is that the list of invitees has grown and now some folks with power but not context have gotten into the room. I feel that their confidence in our team has dropped due to them seeing "dumb" questions (things like, "hey, does anyone know how such-n-such-critical-component work?" with responses like, "doesn't it do blah?" "no, it does blah blah ... I think. Don't recall."

We previously had a public room and a private room for chat. But with the private room having some important folks now, I'm not sure we can just kick them out. We might need a private-private room.

Thanks for the question, it reminded me to have this conversation with my manager (as we would want her in the private-private room, but want her blessing and input - maybe she is fine telling the others to leave).