Anyone who views a public web map can make changes to it. They can add layers, add new features and so on. However, they will not be able to save the web map and overwrite the original version. They can, however, do a "Save as" and save it as their own map.
Presently, there is nothing you can do within the ArcGIS.com map viewer to change the editable nature of the map. We have considered allowing an author to make a map read only, but this is not currently scheduled on our release plan. You might consider embedding the map in another web page, like this:
http://mikesarcgismaps.blogspot.com/
Or, you can use an application template and deploy the application on your web server (if you have one).
Both the embed and create template options are available under the Share menu on the Map Viewer.
Thanks,
Mike
My issue is that other users are able to make edits to the map and layer when they view the map. Is this right?
Tim,
You have to host a feature service on your GIS server in order for everyone to be able to edit the features. Map Notes do not support this type of editing.
Thanks,
Mike