This happens all the time. My sisters and I have/had completely different parenting styles. It did make for some uneasy family gatherings.
I don't know if your daughters' situation will create a permanent rift, but you can take hope in my family situation. Now that our kids are older, there is far less judgment and criticism. We get along fine now. I'm very close to one sister in particular even though we were very different in parenting styles.
You might have a talk with your daughters and explain that every parent has the right to choose how to raise her child without being judged or criticized. Whose way is better? The proof of the pudding is in the eating. As long as their kids grow up intact, healthy, happy and productive, a mother can say she did a good job. Kids are different and require different parenting styles. God loves diversity.
Are they religious? If so, they shouldn't pass judgments upon each other but just love.
I do not agree with some of my sisters' lifestyles or parenting at all. AT ALL. I feel that some are damaging and indulgent. That was hard to bear. However, I love my sisters. They have a right to live as they choose. I'm not perfect either and don't profess that my way is the best.
Please explain to your daughters not to let these issues come between them. One day, they may need each other should divorce or death come into the picture. Certainly, empty nest will come and they'll want each other then.
As for the current situation, maybe it's a good thing for your daughter not to babysit for her sister...or have one daughter pay your other daughter to babysit. Remind your babysitting daughter to love her nephew and niece and not take out her frustrations on them because of the kids' parents. And that she can be a good guiding force in her nephews and nieces' lives with fairness and consistency and love.
And remind your daughters that they love each other and nothing should come between that. Life is for loving.