Bookmum, as explained above, the privately educated children won't count in one of the key school accountability measures - Progress8.
As a result, having 'properly stretching' targets for them doesn't really matter from the school's point of view, as long as their 'raw results' will look OK.
So a privately educated child getting, say, 6s or 7s may be absolutely fine from the school's point of view, as they will count positively in their ' Got good GCSEs including English and Maths' stats. However, a child of the same 'basic ability' with very high SATs results from primary may be given targets of 8s, and receive lots of extra support should they be falling below them.
Of course in most cases it won't make a difference to the actual results obtained - the privately educated child will be given appropriate targets based on in-school testing, and will be taught well, and will attain those targets. This will be particularly the case because schools with many privately-educated pupils will often be desirable selective schools (in such schools, a quarter or more of pupils may come from the private sector at 11) and those schools will by definition tend to have high raw results for all.
It's just that, at the margins, a school with necessarily limited resources in terms of time, teachers and money, may well provide that slight extra push to those children who will make the most difference to their particular accountability measures, and that may slightly disadvantage those children who are underperforming at secondary for whatever reason, but where this has no baseline to be tracked against.