In my case the private practitioner has sent the report to the LA and is awaiting the "go ahead" from them to forward it to me
Not for ECHP but for other reasons of concern, I've had private Ed Psych and SaLT reports done which I've paid for and they were sent to me with a copy that I "might" wish to give to school who could then implement the recommendations.
From that, I'd guess the person/body who commissions and pays for the report would have ownership of it.
I'm concerned that if the LA were unhappy, they could ask for redraft before it is shared with me
In your shoes, when you do get it, I'd ask for confirmation in writing of any parts of the report that had been redrafted with an explanation of why that happened and what it originally said.
Also, frankly, I am suspicious of the relationship between this practitioner and the LA, as the LA refused to consider a practictioner without a waiting list and made us wait a few months for this assessment
That would also concern me, particularly if the report had been rewritten and I'd be likely to ask the LEA's justification for insisting on that particular professional.
However, it could be that the private practitioner also works for the NHS so the LA will accept their report whereas they don't "believe" reports from totally independent professionals.
There's a really odd thing about reports, may be just my own experience or may be common, i don't know. Our GP gives no credence to any reports unless they are done on the NHS, so if the GP is asked if a patient has x condition and there's no NHS report to say so, but there is a private one, then they will say they have no evidence of that patient having that condition.
To complicate things even further, sometimes the long NHS waitlist for a practitioner is because they only work one day a week for the NHS and the other 4 days for their own private practice.