Should we tell MIL with dementia that she has been sent home on hospice?

Hospice does not mean death. Hospice means better and more attention/visits to the person.

If Hospice has become a "dirty word" in your home, just tell her it's Home Health Care.

I wouldn’t tell her. My mom graduated from hospice twice in her 2.5 years in the NH. I never told her anything either time and the difference in her care wasn’t obvious to her. Or me either actually. I learned to try to buffer anything that would get her nervous or anxious or sad.

I would see no need to tell her. What difference really would it make?

No. Why would you? My friend, who chose the Hospice facility & was the most upbeat person I ever came to know in my 64 years, sank into a deep depression when it was clearly time to go there. She lasted maybe 10 days.

Is she getting good, appropriate care?

Is she comfortable?

Do you think that telling her that she is on Hospice would accomplish something positive?

I would leave it alone for now. Sometimes, folks on Hospice improve to the extent that they "graduate" from Hospice and curative, non-palliative treatments can be tried.

If at that time her mind is clear, you can have a talk with her about the pros and cons of doing Hospice care again. I wouldn't bring it up right now.

No need to tell her but please tell the Hospice team that mom does not know she is on Hospice.
Was the "stomach bug" ever diagnosed? Was that the reason for Hospice?
There will also be a CNA that will come in a few times a week to help bathe her and order supplies.
A social worker and chaplain may also visit.
You do not need to accept the services of the CNA, social worker or chaplain. The only one that is required to make visits is the nurse.
But having the CNA is a great help. And it does make it easy to keep the proper supplies on hand.

If you tell her just tell her that this is the way to get her more care, special program and etc.

No need to tell her. If u feel you need to say anything just say the doctor ordered Homecare. (Where I live Homecare also handles Hospice) I will be nice for you because u get an aid maybe 3x a week and some supplies like Depends.

Just a by-the-by...legally speaking, IF she is truly oriented, she has the right to know; the hospice staff also can not be told to lie. However, there is no reason to make an announcement. I would use the gentler, fairly accurate terms suggested, but be aware that direct questions from her... should be answered truthfully, and then be ready to share all the positives that you see.

You and only you can make that call.
You should know her better than anyone and you probably know what she would want. Some people want to know and some don't. Now if she isn't going to remember what you tell her than it would be no point in telling her. But if she knows what is going on, then she has the right to know, If you think she would want to know.