Hep A can basically be gotten anywhere... contaminated drinking water, shellfish caught in contaminated water, feces cells from not washing hands and then getting it in your mouth, etc. The Hep A vaccine helps prevent this from happening. Someone who gets it can get it for a short time and after they recover it doesn't come back.
Hep B and C are chronic and those most at risk to get it are teens and adults, and they're viruses. They're passed the same way as HIV is passed (bodily fluids exchange) but much more easily. A mother delivering her baby can pass it to the baby during delivery. There are other ways too, like using unsterilized equipment at nail salons, changing dirty diapers, getting a tattoo with dirty needles, blood transfusions, etc. There's a vaccine for Hep B but not C.
It's usually recommended that family members in the same household receive vaccinations. The person who has Hep should be on a healthy diet and often will be taking medication.
I don't think that this person's medical status should have been made public to you unless you were living with them. Simply being babysat didn't/doesn't put your children at risk there. And frankly, your choice not to vaccinate your children on the recommended schedule and your choice not to participate in getting them specific vaccinations does NOT obligate anyone else to disclose their personal medical conditions. It's up to YOU to protect your children. It's up to YOU to figure these things out. If you want your children protected as much as is possible without damaging familial relationships, you may want to re-examine your position on Hep vaccines.