It is said that gossip is a simple rumor of personal nature.
Perhaps what it says is that "gossip is a simple rumour of a personal nature". Matters "of a personal nature" are matters that are private to the individual, especially regarding relationships, health, money, family etc. Or as you say, 'private type of stuff'.
This is not exactly the same as someone's "personal nature" in the sense of their character or personality, although of course those aspects will often come into the gossip.
not a teacher
Student or Learner