There’s a good (but not obvious) workaround in iOS. You can set Emergency Bypass on contacts for both SMS and phone calls, which allow those contacts to bypass Do Not Disturb.
First, ensure that you have a PagerDuty contact set up in your contacts list, and add the phone numbers that PagerDuty calls and texts from to that contact.
Second, follow the instructions here to enable Emergency Bypass on that contact’s ring tone and contact tone. https://www.tekrevue.com/tip/iphone-emergency-bypass/
Then you want to ensure that you have PagerDuty’s notification rules configured to notify you via text and/or phone call. My preference is:
Immediately: send a push notification via the app. This will use whatever sound you have configured in PagerDuty, but will honor Do Not Disturb.
After 1 minute (and again after 2 minutes): text me via SMS on my phone. I have a custom text tone for the PagerDuty contact which is a quick, subtle sound and vibration. This will bypass Do Not Disturb, but if it goes off sometime where silence is preferred, it’s really subtle. You could even set this to no sound, JUST vibrate. So you can get a very subtle notification even in Do Not Disturb.
After 3 minutes (and again after 4 minutes): call me on my phone. I have a custom ring tone for the PagerDuty contact which is a distinct sound and vibration. This will bypass Do Not Disturb. You could even set this to no sound, JUST vibrate. So you can get a very subtle (or not subtle) notification even in Do Not Disturb.