By default, Shotwell stores all metadata only in its database. There's an option in the Preferences to enable writing metadata into files (it's on the first tab in Preferences in Metadata section). This way you get tags in sync.
Edit: Expanding on the Danny's answer, you could also sync Shotwell's database directly if you want to sync everything. This will however work only if your photos are stored in the same location on all computers where you want to sync the database.
This can theoretically lead to database corruption or data loss if you have both Dropbox and Shotwell running on multiple computers (and therefore there's a risk of edit conflicts).
Database is stored in ~/.shotwell/data/photo.db – either symlink it or move it into Dropbox (and then symlink it back) – I will assume you use ~/Dropbox directory.
Run this on your laptop (where the database is most up-to-date):
mv ~/.shotwell/data/photo.db ~/Dropbox/photo.db
ln -s ~/Dropbox/photo.db ~/.shotwell/data/photo.db
Now on your desktop, backup or remove the local database:
mv ~/.shotwell/data/photo.db ~/.shotwell/data/photo.db.original
(or to remove: rm ~/.shotwell/data/photo.db)
And again, symlink the synchronised database from Dropbox to the original location:
ln -s ~/Dropbox/photo.db ~/.shotwell/data/photo.db