We opted to send DS1 private at 11, and DS2 will follow the same path from September. We live in a grammar area, so our choice of state options is already a bit different to the norm.
DS1 is highly academic, but lacked some of the social skills - our rationale for sending him private was far less to do with academic achievement (he is on track for excellent GCSE grades, but I doubt there would be any material difference in outcome compared with what he could have achieved in the state system). However, we did think that the private option might just give him a bit more opportunity to grow and develop his social skills, particularly with the co-curricular opportunities. Personally, I didn't much like the local grammar which he would otherwise have gone to, so opting out of state was an easier decision for me. I also will confess that I thought it would probably do DS1 the world of good to be exposed to children whose parents work in industries and job roles which might lead to better opportunities in the future - his 'black book' of contacts if you like. There is also quite a strong network of former students who might be able to provide different opportunities and contacts.
For DS2, the decision was somewhat different. He missed out on the 11+ (narrowly) and also on the entrance exam for his brother's school, but passed a couple of other schools' entrance exams. He is a much less academic child, and also rather more sensitive than his brother. So for him, the advantages of private school are more likely to focus on the smaller class sizes, less opportunity to drift (which I suspect would have been a likely outcome in the local 'secondary modern' state options), and a nurturing pastoral care team. We'll find out more once he starts in September, and hopefully the choice will prove to have been a good one.
Would I do it again? Well, yes, and I am doing it again for DS2! It's a huge financial commitment so it is difficult to say if it's 'worth it' - I don't think we'll ever see a clear £ benefit that we can point at and say that it was directly linked to a private education. But I am grateful that I can make the choice.