I think with young children it's possibly quite feasible but as they get older children throw up more and more parenting challenges potentially, and the world outside the home affects them far more than it does pre school children. The potential for conflict/ different approaches will increase. It's one thing to compromise with the child's father, but why would you want to do that for a friend who has no relation to your child?
Similarly, young children tend to muck in and play together quite well, unless one has particular needs. By around 10 they will have preferences and have developed their own personalities. How easy will it be to negotiate times / spaces when they need to be together in the house.
I also wouldn't want to be the one living in a house that wasn't my own. You'll always have the "power" in that regard. If you meet a romantic partner, does she have to move 90 minutes away and take her children out of school. Or give notice to tenants, sell her place and wait to buy in your area? That's a huge disadvantage potentially.
What if you get on each other's nerves? Living together isn't like a nice day out or a holiday. It is easy to get along in those circumstances. Also potentially easier for two childfree women who house share, have their own space and are free to come and go as they wish. What you seem to be talking about is co-parenting. You'll be dependent on the good will of each other to agree on you each having time off for nights out, hobbies etc. Again more readily negotiated (though not always that easy either) by partners who share parental responsibility. What if you (or her) meet a man and want to start seeing them regularly, but the other doesn't?
Financially, I'm not sure how it would work. Should she contribute to house repairs in your house? Share the purchase of new appliances you both use? How easy will it be to divide up the outgoings fairly? Maybe she likes the heating on more than you, or uses more water etc. These things cause conflict in many flat/ house shares as it is.
As someone who lodged with several other women in my 20s (several different places), I would never go back to living in someone else's house again. It wasn't all bad, but the day I bought my own place was a very special one!
I would also prefer to keep my best friend just as that. We have been good friends since 16 and shared all aspects of our lives, been on holidays together etc. But I can see we don't 100% have the same views on everything (of course not) and sharing parenting would never have worked (not that it could have been an option as hers are quite a bit older).
Just to counter balance the majority positive view on here!