What would be your understanding if told you were 'self aware'?

[MrsSeanBean]

A colleague uses this phrase often and it leaves me rather confused. As in: 'she is very self aware', also 'he has no self awareness whatsoever'.

....?

I think it means being conscious of one's behaviour - or of how one's conversation/actions impact on others. Having no self-awareness means being impervious as to whether what you say could be offensive or rude, or makes you look like an idiot, etc. But being very self-aware I would take to mean very careful, deliberate, possibly awkward. Does that make sense?