Well, we do support a PagerDuty integration which will of course do the things you’d expect PD to do. But then you’d have to have & pay for a PagerDuty account.
We are not intending to duplicate the PagerDuty featureset (eg: re-escalation if the incident is acknowledged but not resolved) so our best advice there is to actually use PagerDuty, or as you are clearly trying to do, let us trigger some of your code to execute whatever business logic you have, via webhook.
If you could describe specifically what escalation/notification features are missing for you, we can advise what we can accomplish (for instance: we can at present send an SMS via email<->SMS gateway, and we can notify you about each new problem within a policy, if you configure the policy’s verbosity level as documented here:
Another possible workaround is to have your webhook receiver live somewhere free and public (eg: heroku - low traffic single-dyno apps are free) and then you can have that receiver which our system can access forward the notification into your system, since you’ll have access to full language capabilities for your app to do HTTP auth or whatever else needs be done to access your internal infrastructure.
We’ll definitely be filing a feature request about the things you’re missing, so please be specific so we can make sure that any future development has a chance of solving your base-level problem(s).