Your English is somewhat broken so this answer is based on my interuptation of it.
Permission to send Broadcast Messages are controllable per-mailbox, and by extension User Templates only. Distribution lists are not involved in a system broadcast message.
The Broadcast Message Administrator allows a person to create or update broadcast messages similar to how Greetings Administrator allows a user to call in and record/change greetings on Call Handlers. These are actions they cannot do within their normal voice mailbox.
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