Our non-Japanese children went the public route, as we were both working and needed to keep it „simple“. One had almost no problem and lots of friends, the other had a horrible experience, especially during Junior High. As s. o. pointed out there are so many factors you cannot control, so it is almost impossible to tell beforehand how it will go. The situation can also change year by year, as classes and teachers change yearly (or in some schools every two years).
I don’t have personal experience with private Elementary schools, but through observation and talks with friends who had, I learned:
a) Your child will likely not develop any friendships in the neighborhood, even if it takes an extracurricular activity locally.
b) Most likely she will have to get up early in order to commute, her school day will thus be quite a bit longer, can be hard on a 6-year old.
c) Children (when starting from Elementary) are not only selected by their intellectual and social abilities, one important point seems to be that the parents have to be 100% supportive of the schools goals (whatever they may be) and much more parent involvement (= usually Mother involvement
) will be expected. This can may make it very hard for working mothers, who in this environment are probably still in the minority.
d) There are (usually) strict dress codes for the mothers, you may have seen groups of women in all navy costumes, navy handbags, shoes etc.., they dress like this for any school meeting.
e) Your child will not meet many children from varied backgrounds, whether this is good or bad I don’t know; it limits your life experience though.
f) There may be no school lunches (prepare bento = additional hassle for mothers and I guess it cannot be just any bento…)
g) If the school is quite small you have a limited choice to find friends and if there is a bully it is harder to avoid him via a class transfer. On the other hand friendships formed, may be deeper and longer lasting than in larger public schools.
h) It should be easier to teach children with similar abilities and backgrounds and where you have the parents full support and engagement. Reading is strongly encouraged, you often see kids reading on trains.
i) To the monthly fees you have to add: Commuting fees, uniforms - winter/summer, coats, hats, shoes, socks, etc., (which are much more costly than e. g. Public high school uniforms) also it seems that many (all?) schools expect yearly „voluntary contributions“ of a few 100thousand Yen.
j) Of course it will easier to get into a good university (or the university associated with that school) as that is the whole point.
k) If you start from Elementary you can bypass the harder and more competitive entrance exams for Private schools from Middle school (usually as then your child typically will start with Juku from grade 3 or 4) and from grade 5, 6 there is very little time for play left, which I find sad, but that is just my foreigner‘s view.
l) Most - but not all - of the private school have great facilities, far better than the public schools.
m) Teachers should be more devoted and don’t get transferred ever 4 to 6 years, as in public schools.
When describing the above, I have a certain picture of „Ojosama/Obochama“ schools in mind, there are probably other private schools out there that are more varied/more progressive.
Apart from the financial side, it depends on your family’s values/the goals you have for your child and her personality which way to go.
Good luck.
) will be expected. This can may make it very hard for working mothers, who in this environment are probably still in the minority.
but they came out wrong as I copied them from outside instead of selecting them from here. 
Lesson learnt.