It is a bit complicated so I apologize for the length. The short version is: if a description is Draft, then both the metadata and any linked digital object will also be hidden from public users. When you publish a description, its metadata and the related digital object will also be visible.
The longer version:
Archival descriptions are currently the only entity in AtoM that have a publication status. There is a setting (described here) that determines the default publication status for new records created in AtoM - when you first install AtoM, this setting is set to "Draft," meaning when you create new descriptions, they are draft by default, until someone publishes them.
The publication status affects the visibility of the description to public users - by default in AtoM's permissions, anonymous (i.e. public) users cannot see Draft records. This means they should not show up in search or browse pages, or even if the public user knows the URL to go directly to the view page of a draft description. However, also by default, anyone who logs in can see draft descriptions.
Digital objects are a bit separate, but they only exist in AtoM when they are linked to a record - right now, users can link digital objects to either archival descriptions, or authority records. (Note: authority records do not currently have a publication status, so they are publicly visible by default)
If a description is set to draft, then the entire record is hidden from public view - and this includes both the metadata and any attached digital object. When published, it is the same - both the description metadata and any linked digital object will now be visible and discoverable in search/browse to public users. By default, public users can see the thumbnail and the reference image of your digital object on the view page of published descriptions, but they cannot access the full-size original upload version unless you make changes to the anonymous group permissions.
There are also other ways in AtoM to hide elements of a description from public users.
For example, the Visible Elements module lets you hide specific metadata fields from public users. For example, if you want to publish a description, but your archivists always use the ISAD Archivists Note field to add private internal information, then you can use the Visible Elements module to make this field hidden from public users (even if the record is published). The effect is global - if you hide the Archivists Note field in the Visible Elements setting, it is hidden in ALL archival descriptions. See:
Similarly, it is possible to use the PREMIS Rights module to hide a digital object linked to a published description from public users - so you can make the metadata visible and let users know that a related digital object exists, without actually showing it to them. This can be useful if there are copyright concerns, or if you have a donor agreement to keep the objects private, etc. See:
I hope this helps - please let me know if you have any further questions, or if you would like me to clarify anything I've written above.