Home Assistant is open source home automation that puts local control and privacy first.
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Home Assistant can be self-installed on ProxMox, Raspberry Pi, or even purchased pre-installed: Home Assistant: Installation
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You could try Rhasspy - https://github.com/rhasspy/rhasspy
Mike is a pretty cool guy. I know the repo is archived, but he might help you if you reach him.
Alternatively, what about a panic button? Both commercial solutions and privacy friendly local solutions exist. I know it’s intrusive and grandma will have to keep it on her person at all times. But… that’s kind of what they’re there for.
I don't have an answer to the direct question, but if she would struggle to use her phone in that situation I also imagine she might struggle to clearly say "Hey Alexa, make a call to Mommy Jim" or whatever. I'm walking at the moment so I don't have the details to hand, but mmwave presence sensors can distinguish between someone standing and lying down. You can also buy Bluetooth wristbands that can detect if the wearer falls. It might be worth figuring out if a fall can be sensed rather than reported.
IDK about a true voice call but you could definitely set something to play a message through a second satellite. You could even make it recurring until snoozed e.g.:
Grandma: "Computer, I need help"
HASS - Trigger automation say (through separate speaker) "Grandma called, she needs help", delay 30, repeat until button is pressed.
The tricky part is teaching it enough permutations of "I need help" that Grandma doesn't have to get the phrasing exactly right in an emergency
A phone seems like an easier solution, but there are VOIP integrations for HA, as well as ways to stream audio through the media player to different devices. I'm not aware of any integration that specifically does paging as you describe, but it would be easy to do with an esp32 board with a speaker.
problem is a phone only works if you have it on you. Grandma falling often is a case where she doesn't have a phone (often grandma in this case is in mental decline and wouldn't think to use the phone if she had it).
A privacy respecting device that can hear "help" and general cries/grunts to get help when needed is what the world really needs. This is much more complex than the ask, but it is the real need here. For me what the OP was asking is enough, I just need to inform a kid in a different room dinner is ready.
Look at this solution. Works even without electricity!
https://www.medienwerkstatt-online.de/lws_wissen/bilder/1062-4.jpg
Go stick your fist in your mouth. That also doesn't need electricity but will improve the conversation.
https://www.percussionplay.com/instruments/talk-tubes/
If communicating between relatively fixed points, this may be more effective
