Once the county acquires guardianship, they will take over meeting her needs for protection and medical care. They will also take over managing all her affairs, so family and others will be blocked out of acccounts or from having any access to information. All her assets will come under the control of the guardian.
But your profile says your sister is MPoA (healthcare proxy) for your Mom... she may need to resign this in order for guardianship to take place. I'm not sure since the rules may vary by state. My family's experience with a court-appointed guardian was positive: they got my very resistant SFIL into a facility (advancing Parkinsons) and took care of him. We were able to visit him freely. They even moved him to a facility closer to our home at our request.
I just wanted to point out that being a guardian does not obligate you to either physically or financially care for someone, if it did there wouldn't be people who do it professionally. You should also be aware that although a good public guardian will try to work with the family they are under no obligation to do so and your opinion about how or where she is cared for will be out of your hands.
She will become a ward of the state. If you refuse to accept her from the hospital, they'll keep her until a nursing home bed becomes available, and then she'll be placed.
But your profile says your sister is MPoA (healthcare proxy) for your Mom... she may need to resign this in order for guardianship to take place. I'm not sure since the rules may vary by state. My family's experience with a court-appointed guardian was positive: they got my very resistant SFIL into a facility (advancing Parkinsons) and took care of him. We were able to visit him freely. They even moved him to a facility closer to our home at our request.