We heard creating a public sharepoint site in E1/E2/E3 plans. But, its exactly straight forward to create sharepoint site for anonymous users in Plan P1. We are not talking about converting a public site to a sharepoint site which Martin defined it here very nicely.
Now, To create the sharepoint site under your P1 plan, please follow the below steps :
1. Goto your public site & add "_layouts/viewlsts.aspx" to it. In other words type "http://<yourpublicsiteurl.com>/_layouts/viewlsts.aspx"
2. Now, hit create and create a site named 'SS'
3. This will sit right under the root.
4. Now, from your public site you can create a link to this url "http://<yourpublicsiteurl.com>/SS".
This is cool & visible for anonymous users. The only issue might be granting anonymous users some more privileges. For example, you might want to allow anonymous users to do a 'Posting' which is by default restricted. You could follow this other post for following that in detail.