Hi, what a difficult dilemma you have. On a personal level this happened in my family: my brother went through the private sector whilst my sisters and I went to the local comp. It caused some resentment as we got older, particularly as my brother didn't even complete his degree, and dropped out.
On a professional level, and I should declare that I am a school placement consultant, in addition to scholarships some private schools offer sibling discounts and may even be able to assist with a further bursary (these are generally means-tested to ensure the right families are getting assistance). Financial assistance might make it possible for you to send both children.
In addition to the comments you have already received, bear in mind on a practical level that if one is private and one state then they will have different holidays, which may not always overlap. Private school holidays are always longer too which may impact on your childcare arrangements.
Alternatively you could consider state education at 11+ and private education at 16+ for both your children, as this would give you time to prepare financially and would remove any likelihood of sibling resentment.
Happy to speak to you if you wish to private message me when we can talk about this in more detail.
Whatever you decide, I wish you good luck.