Can we omit a duplicated preposition? [duplicate]

The answer is that the second on is not needed, and may be deleted.
But that doesn't have anything to do with prepositions, really.
It works for just about any kind of word. Under the correct conditions, at least.

The name of the syntactic rule involved is Conjunction Reduction. It applies only to the second and following clauses in a compound sentence connected by a coordinate conjunction like and, or, but.

What it does is delete material -- adjectives, nouns, prepositions, you name it -- that's duplicated in the second and following clauses. Because it is duplicated, and therefore is redundant.

If you start with

• There is a bad effect on A or there is a bad effect on B.

you can delete there is a bad effect on, which is repeated in both clauses,

• There is a bad effect on A or B.

or you can delete less of the repeated material,

• There is a bad effect on A or on B. (deleting there is a bad effect)
• There is a bad effect on A or a bad effect on B. (deleting there is)

I would change the sequence of the two words "either on" to "on either":

There is a bad effect on either human health or the environment.

By changing the word order to place "on" in front of "either", the question of whether to repeat "on" is rendered moot: the word then clearly applies to both "human health" and "the environment".