In the server admin console, go to Database Server, the Security tab. Check the option labeled "List only the database each user is authorized to access." If it's checked, then FMPro will ask users who they are so that FMS can check to see which databases the credentials allow the user to access. If it's unchecked, then users will see all databases on the server.
As for why the database won't show up, check all of the privilege sets in Manage>Security. Make sure Access via FileMaker Network (fmapp) is enabled for each privilege set that you would like to have access to the shared file.
Thank you so much! That fixed the problem!