Let’s say you have a private group called foobar set up like this:
“Who can see this group?” → “Group owners, members and admins”
“Who can message this group?” → “Only group members, moderators and admins”
If a member of the group tries to send a message to it by creating a new message and typing foobar in the “Add user…” field, they’ll fail. The system will not list foobar as a selectable choice.
If you change “Who can see this group?” to “Everyone”, you’ll be able to select the group and send a message to them.
Note that AFAICT, it’s solely a UI issue. Even when the group’s visibility was limited a member of the group could send a message by going to the group’s page (e.g., https://connect.amicalnet.org/groups/foobar) and using the “Message” button available there.