Had a standard non-HDHP medical insurance plan with both and my copay was $50 for my first and $100 for my second. Different insurances, slightly different co-pays. But it really wasn't a lot. I had my kids at a stand alone birth center, not a hospital. I was transferred to a hospital a few hours after my first was born. I was hemorrhaging after delivery, so they needed to check if I need a blood transfusion. And the midwife, while trained to do stitches, thought my tear was rather messy and preferred to have a surgeon do it. (No transfusion was needed, I was back at the birth center with my baby and husband after 4 hours in the hospital. That bill was maybe another $100 in co-pays. With both, I was measuring small, and additional tests and ultrasounds were ordered. Each ultrasound was another $50-$100. All in all, probably less than $500 both times.
I always think responses to these threads tend the be all over the place because there is such a huge range and it really depends on the insurance you have, and personal choices in birth, or how your labor and delivery proceed, and your care provider's recommendations.
Let's say I had zero insurance and it was all to be out of pocket. Labor for my first started on a Sunday night, I didn't have my baby until Tuesday morning, zero medical intervention was required to bring baby into the world. The birth center global fee for the pre- and post-natal care, and the delivery, would have been $5K.
I am almost certain that if I were in a hospital, an OB would have recommended pitocin to speed things along at some point in my long labor. And then if I had the pitocin and had infamous pitocin contractions, I may have caved and gotten an epidural. And then if the epidural lowered my already low blood pressure like it often does, baby's heart rate might slow. And if baby seemed in distress, than the OB might call for a c-section. This would have easily been $20K+ without insurance.
Of course, some people go to the hospital and everything moves along quite smoothly without intervention. So it's a very personal choice, and it is difficult to predict how things will go.