Consider a chat room, in which N tv-stations are negotiating with the
Olympic committee for purchasing the broadcast rights to the different
sports in the next olympic games. Each station's delegation consists of
one member.
As often happens with these fictional negotiations, at any point during
the negotiations, tv-stations may start to swap and trade amongst each
other with the broadcast rights of sports they've already secured. These
unofficial negotiations happen in private, randomly and un-structured.
To accomplish this using only a sidebar mechanism, in addition to the
main conference (N+1 participants), we would need a full mesh of N x N
sidebars at least (for each pair of tv-stations) so that the delegates
could each send a private communication to each other whenever they feel
the urge.
With our mechanism, you don't have to have any sidebars to enable one
delegate to send a private communication to another delegate.