After stroke Mother can't move right side, but the nursing facility is releasing her. Sister and I don't know where to start. Advice?

You will need to tell the nursing facility that discharge home is unsafe. Be certain to use the exact words of "unsafe discharge". Tell them that placement will be necessary for Mom at this time, and their social worker should begin working on that, that you are not living in that area, and not in a position to take your mother to your home, and that there are no other options. Then your mother will be placed. Yes, what funds she has will be used. If Mom has no assets then the places for her placement will be limited, and she will eventually fall to having to accept medicaid. Good luck. They figure this out all very well and very quickly when there are no siblings at all. But when there are siblings and children they will try to get you to assume all responsibility. So glad ways and wane has been there before you and has some valuable numbers below for you.
If your sister attempts to take Mom home then she will set everything way back and have to handle things on her own. So try to get this into the hands of the social worker for real hands on help.

I had to figure all this out in California.
Call 2-1-1 and ask them what local services are available to you. They provide local, curated referrals.

Call Caregiver Action Network at 855-227-3640 and ask questions. 8:00 AM – 7:00 PM ET. They are a nonprofit and act as aging life consultants FOR FREE.

Call the Eldercare Locator at 800-677-1116 Monday - Friday, 9am - 8pm ET. If you have trouble getting through, go to https://eldercare.acl.gov/Public/index.aspx and search for your local office number by putting in your zip code. This is a government service (local Area Agency on Aging) which provides geriatric care consulting for FREE. They can offer information about transportation options and in-home care support (bathing, dressing, sitter services, preparing meals), home modification and caregiver support/respite. They can also inform you about other local caregiving services, like those provided under the National Family Caregiver Support Program.

You can help by giving your sister staunch moral support. She may encounter a lot of emotional blackmail and hard selling. Help her keep her eyes fixed on what your mother's real needs are and the fact that she can't supply them.

Your sister must refuse to take her home. Your mother CAN get medi-Cal & stay in a nursing home. You aren’t with your mother right, your sister is? She needs to work with the social worker/discharge planner at the facility she is and insist that it’s an unsafe discharge. They can help her with medi-cal eligibility. Or your sister can go down to social services in her county to apply and speak to an eligibility worker. All of your moms income will go to the nursing home and medi-cal will pay the remainder of the bill.