You will need the Full Write Service Plans permission.
Navigate to the service plan you want to update and click on the colored box next to the header at the top. Then choose either Public or Private depending on whether you want to allow public sharing or not.
If you choose Public, a pop up will appear containing options which control exactly what is shared. You can include as much or as little in the public version of the service plan.
Once you are done, click on the Enable button to enable public sharing. Once you do, a "Share" box will appear in the upper right with sharing options as well as a button which takes you to the public page for the service plan. You do NOT need a login to access that page.
Clicking on the cog icon to the right of the "Share" header will open the share options set previously so you can update what information is included in the public page.
Helpful Tips
This setting only controls whether you make available a public link where anyone can view the order of service outside of the system. Scheduled volunteers can always access the service plan regardless of this setting.
The public version of the service plan will not be listed anywhere online. It would be up to you to share the link to your congregation as appropriate.