DGS is a lovely state school, however in my opinion it doesn't compare to private, the benefits of sending your child to a private school definitely is the quality of education.
I attended DGS briefly in between private schools (my family moved back to south ox and they had heard how great the school was) unfortunately it is the same as most other state schools, the classes are too big, the teachers are not given enough freedom within the syllabus leaving lessons un-engaging, and not enough is done to safeguard kids against bullying.
In terms of private schools in the area
St Helens; very pushy, results driven, great if your child thrives under pressure. Most of the girls are lovely and go on to attend decent universities. Runs a very efficient school bus service.
OHS; Results driven, not as pushy, a real city school which is nice if you live in or near oxford, not so great if your a bit further out as getting in and out might be a hassle.
St. Edwards; Fully rounded, welcoming school (starts at 13 which might be a problem) co-ed, great if boarding, avoid if you're only looking for day as the day pupils start at 9am and finish at 9pm.
Leckford Place; Feeder school for D'overbrooks sixth form, small classes, excellent teaching, still a relatively new school, co-ed. Still the same problems with it being in oxford, but this school encourages independence meaning most children get themselves home, i believe there is also a bus service.
Headington; Very sporty, good drama dept, girls that I've known go through Headington have gone to some mediate to good universities. Recommend this as a day school as the boarding is sub-par.
Hope this helps.