As @Dog says, I haven’t seen anyone go after another member via flags. We do occasionally get people who abuse the flag system, by flagging every answer they don’t like or don’t agree with (even though they are well within the guidelines). Those folks will get a PM about using the flagging system appropriately.
We do take notice of ‘pissing contests’, and will intervene if necessary. But the few we’ve had have all been brought to our attention in other ways.