I too think the private school's results are probably correct as they use standardised testing, typically marked by GL. Whereas state schools make it up as they go along often just use teacher assessment.
DSs levels leapt into the stratosphere upon moving to a private school at the end of KS1 (I had thought he was under-performing), the state school had been suppressing his levels for value add purposes, undoubtedly he would have made expected outstanding progress in KS2, and the school would have done a fantastic job in bringing on my DS. Of course, the private school did try to take some of the glory but even they knew they could not get away with it. When I told the headmaster what levels DS had been on, he looked at me tongue-in-cheek and said 'oh perhaps he has made a lot of progress here', no I said 'I think it was a work of fiction', he laughed and said 'I think it was.' Just to give some idea: reading 5 sub-levels in 6 months, writing 5 sub-levels over a year and maths 3 sub-levels in 6 months.
What I cannot understand in your case, is how the state school would have thought they could get away with it. Most of the testing in yr 6 is externally marked and unless they were going to fraudulently alter the scriptskeep their fingers crossed, your child was going to make no progress in yr 6, so Ofsted would have spotted that.
In my case, I also have a set of school reports that do not match the levels contained in the official school records, although I have had great difficulty unsurprisingly in getting hold of the records to try and work out exactly what went on.
I think the children undoubtedly suffer in situations like these and I think the biggest single change Ofsted/Gove could make would be to stamp this sort of thing out.
The more you look around the more widespread you realise this sort of thing is. It is fraud.