Have you checked out "How to Create New Presence and Push Notification Devices" in the documentation? You can send text notifications all within the hub - nothing extra is needed except installing the Hubitat app on your phone
My phone is listed as a device and is running HE .. My rules said to send text to it, but nothing happens. The only notification I get is if I open HE and look at notifications within the app.
I'd rather have it text me. Something isn't working.
SMS (what most people think of as “text messages” in the traditional sense) can’t be done without an external service like Twilio.
The Hubitat mobile app can send push notifications, as can external apps like Pushover as @oldcomputerwiz mentioned (with a lot more customization/flexibility than the built-in Hubitat app’s push notifications.
I followed this doc originally when I setup HE the first time and it shows my iPhone in devices. My automation rules show my iPhone and I use the mobile HE on my iPhone. For some reason, I get no notifications. That's when I decided to research and they suggested going down the Twilio route. I also check my message settings on iPhone and everything looks normal.
In your iPhone's Settings area, check the permission-related settings for the Hubitat app (and its notifications in particular) to make sure the app can do everything you want it to.
Sometimes, an app's default settings can be overly restrictive (especially with Apple stuff).
Please note that there are 2 apps in the Apps Store. I'm pretty sure the one you want is Hubitat Elevation, not Hubitat; at least that's the one that works for me.