Update: We've released a new feature where message senders can now assign themselves conversations, freeing up admins and moderators to focus on other tasks.
The feature works on a campaign by campaign basis and is optional. To turn it on, an Admin or Moderator would just have to check the box below your open / close times.
The message sender will see this screen if they haven't been assigned any conversations yet:
They will see this screen if they've finished up their initial assignment and want to send more:
Clicking "Request More Conversations" will assign up to 100 conversations from those in the "Unassigned" bucket.
For more information see our administrator article.