I went to a single sex secondary school and sixth form (neither of which was private) after a mixed primary school and honestly, I think my secondary school experience was a lot better because of this. It was evident that many girls were able to speak up in class and ask/answer questions more confidently and were very focussed on their studies. We were also able to have more open discussions in class than we perhaps may have if there had been boys around us, partly due to the awkward nature of certain topics and also partly due to the lack of fear of being labelled a 'nerd' or anything silly like that - if anything, it was seen as highly positive by most girls to be driven in regards to your academics. Another thing I noticed was that when it came to sixth form, there was an incredibly high number of girls studying STEM subjects - about 100/220 studying Chemistry and over 100 studying Maths! - whereas in mixed settings, it has been said that girls are not encouraged as much as boys to study these typically male-dominated subjects. Additionally, there was significantly less 'boy drama' between girls than there usually seems to be at mixed schools, so our friendships really flourished as a result.
However, some may think that sending your child to a single-sex school rather than a mixed school will result in the child being 'awkward' in social encounters with those of the opposite gender - whilst this can be true for some people, it is usually not the case. In fact, although a large percentage of the girls at my school were sent there by their parents for religious/cultural reasons, there were also many girls there who had no such issues. If interacting with the opposite gender is such a big concern of yours, you could always just ensure that your child goes to mixed clubs/has friends of the opposite gender out of school.