You get what you get, basically. Circumcision is the type of medical procedure about which they say in med school "watch one, do one, teach one." Ob/gyns do them all the time -- as do some pediatricians -- but there's no color-coded "cut here on the dotted line," so how much vs. how little the doctor cuts off is a matter of personal style and preference/how they were taught. Some doctors cut so much that there's not enough skin to allow for a full erection and there can be bowing; other doctors cut so little that you might not know the boy is circumcised if you've never seen what an uncircumcised baby boy looks like, and there can be issues with stuff getting trapped under the remnant foreskin.
Bottom line -- you need to ask more questions. There's no rush that it has to be done the day after birth or the day before discharge. You have every right to ask exactly who might do it, what their experience is, what their style is, etc. Or you can wait and find a mohel who's also an MD or go to a pediatric urologist who specializes in the baby boy penis anyway (unlike an obstetrician who by definition is a surgeon for adult women).