Do you open the file with their password or yours when you test it?
At what point does the error message appear? Do they get the hosted file open and then get this message or are they unable to open the hosted file from the server?
This particular file is setup to automatically log in at the moment. They get the error message as soon as the first layout appears. They are able to open and navigate the hosted file.
Here are some more details:
1. I currently have FM Server installed on my local machine.
2. The database file is saved on my local machine.
3. When I move the database to a network location, share it via FM Server, and then try to open it I get the same error message the other users get when they try to open the file saved on my machine.
That sounds like a file reference that is valid as a file path to a file on your local machine that is not valid from the context of a different machine--such as when others connect to the copy hosted on your machine or when you move the file to the server for hosting.
Normally, an external file reference error dialog names the file that it can't find and the wording is different or I'd suggest opening Manage | External Data Sources and check for a file reference that is specific to your location.
I'm not fully conversant on the details for setting up ODBC connections, but that does seem like a prime suspect here. Are you using Server or Server Advanced? Server Advanced offers more support for ODBC than Server so you may need to check the product specs for Server and Server Advanced vs the details in how you are accessing via ODBC to see if that sheds some light on the issue.