I'm in Trafford. I have twins about to start Year 5. They are at a (very good) state school and everyone (without exception) that I know well enough to discuss it is tutoring if their children are sitting the exams. I was advised to put their names down with a well regarded tutor about 3 years ago. When you say the tutor business locally is booming, you have no idea!!
As someone up thread said, the local prep schools prepare for the entrance exams quite aggressively which state schools aren't allowed to do, so the tutoring is levelling the playing field a little.
As far as I know, there are no "stats" about how many children get in generally from state schools. Certainly at my children's school it can vary from year to year. As far as I know, roughly 26 or 27 girls take the exam most years - 8 got in to AGGS 2 years ago, 22 got in last year. They do say roughly 40-50% "pass" but that means a place at a local grammar (i.e. Sale, Altrincham, St Ambrose - not just the Altrincham grammars). Also bear in mind that unless you're catholic, there is very little chance of getting in Loretto or St Ambrose. There have been no non-Catholic admissions to Loretto for a long time (as far as I know), there were a handful at St Ambrose 2 or 3 years ago but not for the last couple of years (unless it was under the sibling rule).
Having said that, the grammars out perform some of the private schools, so children that previously would have gone to say Manchester Grammar, Withington Girls, now take places at the Altrincham Grammars. As a result, the competition is massively fierce for the grammars (I think 1800 took the exam for 180 places at AGGS last year) so moving to Trafford is absolutely no guarantee of getting in. But, the upshot is that lots of children who would previously have got grammar school places are going to the local secondaries (Wellington etc) so their results are increasing massively too.
We moved from a different area before our children started school - education being one of the reasons, but I also didn't realise quite how competitive it is. Not just for school places, but because so many families live here, there is competition / waiting lists for absolutely everything (like Brownies / swimming lessons etc!). Most of the time, I don't regret it because I like living here, it has worked out for us with neighbours etc, but would could have a similar sized house for £100k less if we lived a couple of miles away. You pay a premium to live within catchment for good primary / grammar schools etc but its definitely not any guarantee of securing a grammar school education these days.
Hope that helps.