It seems to me that if you replace "whom other" with "that" you end up with a similar meaning:

    people that people bully bully people

I've found it fairly common to hear people drop the "that" when speaking - so in this case:

    people people bully...

the first people refers to the people being bullied while the second people refers to the people doing the bullying. It's a bit clearer if you replace some of the nouns - for instance:

    dogs people bully bully cats

which I believe is a perfectly legal English sentence.