There's an important distinction that needs to be made: Accessing a file vs Downloading/exporting a file...
The Dropbox app and other 3rd party apps that use the Dropbox API allow you to "Access" files in their cloud location. Everything you do to this file is synced (assuming the app is well made and working correctly). Essentially the file is being downloaded in the background so you can open and work on the file and then reuploaded in the background when you are done (ie syncing). I use apps like Estrong File Explorer and several note taking apps that use the Dropbox API and every change I make to a file is synced when I save the file.
Downloading/exporting the file is different. In the Dropbox app you can "Export" a file which means download it to a local directory on your device. Apps like DocsToGo don't support the Dropbox API so there is no way to directly access Dropbox files from within the app. In cases like this you have to download the file to your device to access it with this app. This downloaded file is not monitored by or synced by Dropbox so you'd need to upload it or sync it manually somehow. There are third party apps that do this, I use DropSync.
TL;DR: If you are editing a file directly in an app that has built in access to Dropbox you don't need to worry syncing, it will happen on it's own. If you are downloading a file from Dropbox to edit it in an app that doesn't access Dropbox directly, like DocsToGo, you'll need to use a 3rd party syncing app like DropSync.
I think this is the scenario that you have... You open a file "example.txt" through Dropbox and it opens a dialog that lets you choose the default app and you choose "XYZyadayada" app and the file opens in that app. You make changes and save them, Dropbox will detect this and will sync the changes. If you skip Dropbox next time and just open up XYZyadayada and see example.txt in the "recent files" list, that is probably a locally cached file. If XYZyadayada truly doesn't communicate with a Dropbox API then that file is saved locally, probably in it's own directory and opening and editing that file will not be detected or synced by Dropbox because it's a copy of the cloud file that is saved locally on your device.