You do not need to turn on the "COVID-19 Exposure Notifications" in your phone's operating system for the TraceTogether App to work in the background. All you need to do is to turn on your phone's Bluetooth, and keep the App open in the background whenever you leave home :)
More info:
The "COVID-19 Exposure Notifications" in your phone's operating system - also known as the Exposure Notification (EN) protocol - is a technology offered by Apple and Google in May 2020 to support public health officials in contact tracing. The EN protocol allows iOS and Android apps to exchange random IDs via Bluetooth, without the apps being in the foreground.
While the EN protocol is a great piece of technology, at the moment, it does not allow public health officials to gain important insights into how COVID-19 is spreading. Given that COVID-19 infections are still on the rise globally, there is much more to learn in order to tackle this healthcare crisis.
Hence, the TraceTogether App does not currently use the EN protocol. Instead, we continue to use and improve the BlueTrace protocol. In the v2.1 app update, our iOS app no longer needs to be kept in the foreground to work — it is now able to exchange signals with Android apps while in the background. Except for very limited circumstances, iOS apps are still unable to exchange signals with other iOS apps if both apps are in the background.
This means that TraceTogether can now cover more than three-quarters of all phone-to-phone encounters: Android-to-Android, Android-to-iOS, and iOS-to-Android. iOS-to-iOS encounters make up much less than a quarter of encounters, based on market share estimates.
Team TraceTogether will continue to improve the App, and we thank you for supporting our contact tracing efforts. Stay safer together, with TraceTogether!