Where does the apostrophy go in 'each others' in the sentence 'The school and pupil respect each others contribution...'
the apostrophe should be here (each other's).

each other in a collective pronoun, representing both the pupils and the teachers, therefore the 'contributions' of the 'pupils and teachers' is their possession. Saxon Genetive rule.
I'm confused by this - allow me to elaborate:

Each other's contributions: this to me says "the contributions beloning to each other", but "each other" appears to be singular. When we imply possession by a plural ending in "s", don't we put the apostrophe after the "s"? i.e. Each others' contributions.

It can be assumed to be plural because "each other" refers to at least one other party, and the speaker.
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"Each other" is not 'plural' as such and does not take an 's'. To test this, you can write out the sentence in its expanded form;

"The teachers and students discuss the contributions of each other."

You would not write;

"The teachers and students discuss the contributions of each others."

By definition, 'each other' must be singular - look up the meaning of 'each'.

There are many contributions (pl) made.

Each person (singular) that makes one (or more) contribution(s) - is still only one person.

"The teachers and students discuss the contributions of each other person."

Carl Barlev
I think it is " each others' ". I agree that the posession of the contributions requires the posessive use of the apostrophy.
It does not matter in the least whether each other is considered a singular or a plural. There is no plural s at the end, so the apostrophe precedes the genitive s: The school and pupil respect each other's contribution.

In the same way:
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how do i use apostrophy