How can I make a folder and its contents automatically owned by two people?
You can solve this problem using group permissions. What you want is to have a file mode creation mask set such that group read, write, and execute permissions are automatically set for each file when a file is created. This can be accomplished using the umask command inside your startup script such as ~/.bash_profile. To set rw permissions on all files, rwx on all directories, that you both create, use

# Set the umask such that user and group can read, write, and execute.
umask u=rwx,g=rwx,o=


Furthermore, you may want to set everything in the Dropbox to have at a minimum rw group permissions for files and rwx permissions for folders. You can accomplish this using

chmod -R g+r,g+w /Users/Shared


for files and use

chmod -R g+x /Users/Shared/FOLDER


for folders. You can view the existing permissions in the terminal using

ls -l /Users/Shared


where the rwx permissions are arranged in order of owner, group, and other.