I have two quick questions re: Private Groups:
I put a site together for a non profit and they want an existing group to be private to be seen only amongst their staff for staff business and is not for public viewing as some stuff may be legal talk or budgetary.
#1 ~ Go to the group you want to make private. Up in the corner, there is an "Options" button. It has a drop down menu, choose "Edit Group". Once you get that screen, you will look where it says "Privacy Settings" and tick the box that says "Group information only (private)".
#2 ~ That means that non members will only see that there is a group there, but they cannot see anything that goes on inside of it. The activity inside of it will NOT show in your activity feed.
Also, a little lower down, you can choose whether to approve all members who join this private group, or you can set it to invitation only.
As to whether it shows up in a Google search, I'm not 100% positive about that. I want to say no, but I'd rather someone else tell you that who knows for sure.