Hi Hugo,

Thank you for the question.

I do not see a simple way to customize it on per group basis.

There are two ways to customize it.
1. Using the filter ‘bp_group_status_message’ or
2. Modifying buddypress/groups/single/home.php in your theme.

Here is an example using the filter.

function buddydev_custom_group_status_message( $message, $group ) {

	if ( 'private' !== $group->status ) {
		return $message;
	}

	// now handle all the cases(invited, pending etc) and modify the $message

	return $message;
}

add_filter( 'bp_group_status_message', 'buddydev_custom_group_status_message', 10, 2 );

Regards
Brajesh