Is there a way to set up additional users so they can see and manage things and the hub from the mobile app, but not receive the text notifications for alerts when the network is armed?
No. The common use case is the opposite, but that is also not yet supported either.
What system owners need to be able to do is define sets of permissions for subservient users.
Notifications ONLY
Read Only
Arm/Disarm SHM
Unlock/Lock Doors
Administrative Only
Full Access
etc…
Where one can define which accesses apply to an individual or group.
Officially, not exactly…
Unofficially, check out the “My Contacts” feature as shown below. I use this and it works great, allowing me to to select exactly which users I want to send Push and/or SMS messages to.
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One caveat is that if you have contact book enabled you won’t get hub offline notifications. Support said my option was to disable the contact book if I wanted offline notifications until they officially re-release contact book
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Greg - Thanks for that information. I had been wondering for a while now how my TWC internet + ST hub could be so reliable and never go offline… 
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It also depends on what you mean by “manage things.”
You may already be aware of this, but just in case not, there’s a very popular third-party app called SmartTiles which allows you to customize what you see. So you can limit it to just specific devices and there is no account level access. And it does not receive the SmartThings notifications. But you can toggle lights on and off and run routines using virtual switches. It’s very useful for situations with housemates, kids, Airbnb renters, etc. – – any situation where you want them to be able to use their phone to change the state of some things in the house but not to be able to delete devices or access other admin functions.
www.smarttiles.click
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Quick, dirty, and effective workaround: Simply disable notifications for the SmartThings app on relevant phones?
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Depends on whether the issue is inconvenience or security. If you don’t want the person to receive the notification because you don’t want them to know the status of the system, then disabling it on their phone won’t help. They can just re-enable it for themselves. But it’s a good suggestion if they are just annoyed about the notifications. 
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Thanks to you both! (ady624). It is just a case of someone annoyed with the notifications.
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I would use pushover for notifications, you can set by device . The app it self is a little limited.