Mailman's "Membership Management" screen shows the subscribers to your list and the various options set for each subscriber. One of these options is the "mod" (moderation) checkbox.
If a subscriber has the "mod" box checked, it simply means that the subscriber will not be able to send messages to the list without the list owner's approval.
For announcement only lists, all member addresses should have the "mod" box checked, except for the e-mail address of the list owner (or other people who should be allowed to post announcements). If you initially create your list as an "announcement only" list, this will happen automatically: the owner will be added as a subscriber who does not have the "mod" checkbox turned on, and everyone else who joins will have the "mod" checkbox set.
This system is very flexible. If you want to allow a few additional people to post to an announcement only list, just uncheck the "mod" checkbox for the extra people.
One thing to note is that if you change the e-mail address of the list owner on Mailman's "General Options" screen, Mailman will not automatically allow the owner's new address to post to an announcement only list. There are two reasons for this. First of all, the new address won't automatically be subscribed as a member of the list. Secondly, even after you do subscribe the new address, that address (as with any new subscriber to an announcement list) will have the "mod" checkbox turned on in "Membership Management", preventing that address from posting messages.
To allow the new owner address to post messages to an announcement only list, subscribe the new address as a normal list member, then uncheck the "mod" checkbox for that address in "Membership Management".