Yes, you'd have to automate the pull for this to work, and then you'd have a way to notify on conflicts so that they are resolved ASAP.
Come to think of it, Dropbox isn't really so great at the notification aspect of this either. It creates a 'conflicted' file but there is no notification that I'm aware of. I have to remind myself to periodically check for conflicts.
Is there a standard-ish git merge driver that resolves all merges with a Dropbox-style "conflicted copy", so you can unconditionally git pull in a cronjob and it never leaves the repo in an unmerged state?