Only moderators have the ability to clear the public chat.
All users have the ability to copy the public chat to a clipboard or download the public chat, however, only moderators have the ability to delete the public chat.
To delete the public chat
Expand the Chat Window  not already visible, then open the Public Chat Menu  located in the upper right of the chat panel.
Select Clear  from the list of options.
It is important to note that clearing the public chat cannot be undone and clears all messages from the chat.