The official Dropbox client preserves file permissions when syncing files between different computers whenever possible (e.g., between Unixes). When changing a file's permissions to executable, this will trigger a sync of the new permissions to all linked PCs. This means that the corresponding metadata is somwhow stored on Dropbox servers.
Does the public API in some way support accessing those permissions? If no, could this be made available in the future? It would be nice for API clients to also preserve Unix permissions, similar to Dropbox itself.