I would use "They face off against each other."

Student or Learner
Are these sentences correct, and what are their definitions?
They face off each other.
They face off with each other.
I would use "They face off against each other."
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.
I think "face off with ..." also works, at least when ice hockey is not the context.
I would use against, but with sounds OK.
You could dispense with the preposition completely by simply saying "John and Terry angrily leapt up from their seats and faced off". It's pretty clear that they were facing each other in that scenario.
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.