Private school (preschool level or not) and whether it will work for your child or not really depends on the school, administration and the teachers that are there just the same as a public school. My son turns 5 in June (We're in Australia where the school year starts at the end of January) and did 1 term at a Montessori preschool (had just turned into a long day care with 2 rooms for ages 2-3.5 & 3.5 to school age and also allowed them to employ lower qualified staff). I wasnt overly impressed so we switched to a private Waldorf/Steiner school the next year. He spent the whole year there (the first half went quite well and he made a really good friend for nearly 2 terms who was a year younger - but then this friend started hitting etc as younger kids do and then DS wanted nothing to do with him). I felt like the last 2 terms he was trying to fit in with the other boys (being more physical and being silly). Looking back now the environment was totally under stimulating for him (he's mechanical in nature and not 'arty' at all).
This year (which he actually qualifies to enter kindy as a 4.5yr old but his fine motor skills are behind so i wasnt comfortable with that) we have paid for him to go to a more elite private school (doesnt cost elite prices though we we're in a small rural university town) transition program (this is a semi play based, semi curriculum based pre-k program). The teacher is probably 45-50, loves her job and is connected with other early educators worldwide which is very obvious in her classroom. Her experience and age really play a role IMO in recognizing my DS's uniqueness and talents. At the start of the year she organized for DS to do LA with the kindy kids while her class does alphabet work and single letter sounds etc. While DS reads at more of a grade 2 level he doesnt tend to read at that level independently yet and still needs some phonics work behind him. He enjoys going to these lessons which is great. Other activities he gets in his classroom are music by an actual music teacher 3x/week, sensory gym 4x/week (like a kindergym program), swimming lessons 1x/week, learning to ride his bike 2x/week (they have a special purpose built bike track at the junior school). They also do some kind of literacy program, art and the rest is filled in with free play (in which the teachers work on special projects with the kids in areas they are interested in).

So far (he's only been going since end of Jan) he's been having a blast and seems to be doing extremely well. As he's the oldest in his class I wouldnt say he has many (if at all) 'true peers' but all the younger kids look up to him and seek his help out about many things as they realize how knowledgeable and helpful he is which really makes our DS feel good about himself.

I dont think we will have the same luck next year in kindy however as the teacher for that grade is quite young (under 30 and possibly more like 25, if i had to guess). She doesnt seem to have the experience with gifted kids and what that means.