I don't think we have a huge number of private schools in Bournemouth.
St Thomas Garnets (Parkwood Rd in Boscombe/Pokesdown) is good. It has a preschool which is excellent, then classes of up to 18 children, co-ed up to end of Yr6. It is Catholic but I know quite a lot of non catholic/non religious children there. Is linked to the Big Little Theatre school (or called something similar to that) which is next door and I think all children get individual drama or something as a result. kids seem very happy, polite and well adjusted. the staff I have come across are lovely. preschool I can highly recommend.
Bournemouth Collegiate is another one, I don't know much about it, day and boarding, co-ed, all the way through to 18. Think the junior department (assume it has a preschool) is over in Parkstone direction near Poole, Senior and boarding is near the clifftop (nice location) in Boscombe/Southbourne area.
I think there is another very small private primary school somewhere Charminster way but I don't know what it is called if it still exists.
I would personally say that for under 11s, St Thomas Garnets is probably the best. If we had the money I would have sent our children there. I know Yr1 is a full year, big bulge year in the area but I think the other years usually have some level of space available. New Headmistress this year, she came from another very good school and is supposed to be excellent.
On the non private school side, the schools are all pretty much overflowing. depending on where you were looking to live obviously affects what would be available to you but they are a mixed bag. Some are great, others not so. some are huge (2 schools are 150 intake at reception!) and I think there are none smaller than 60 intake, quite a lot are now 90 intake.
hope that helps a bit - feel free to PM me if you want any other info about the area.