The terms "Elementary School" and "Middle School" are usually a tip-off that a student attends a school in a public school district.

Catholic and other sectarian schools usually have "Catholic", "Christian", "Bible", "Jewish", etc., in the name. Not to mention the paragraph-long saint name at the beginning of Catholic schools, e.g., St. John the Evangelist, Our Lady of the Seven Sorrows, North American Martyrs, etc. I always loved the abbreviations that children use for schools like that. "You go to Seven Sorrows? I'm at SJE, but my brother is a Martyr."

Private schools seldom have to differentiate among levels -- elementary, middle school, junior high -- as formally as public schools do, because they're not part of a chain, even if they're sometimes franchises. They'll range up to 8th grade (though sometimes only to 6th), at which time their students transfer to private or public high schools. The tip-offs for me are names like "The [whatever] School", "The [whatever] School for [something or other]", "The School of [whatever]", "The New School of [whatever]", the words Montessori or Waldorf anywhere, and anything called a Learning Center or an Academy. Any time a school's name begins with "The", I immediately think "private" until I learn otherwise.

If you want to know for sure, you have to look it up! :)

Regards--
Mary Anne