Peterson is basically suggesting self-reflection as a way to know oneself, which is a fair enough point. Humans have sophisticated capacities to self-reflect and be self-aware, which is a first step towards knowing ourselves. We can also enter in dialogue with others and integrate their perspectives on ourselves into the image we cultivated of ourselves. That way, we can try to counteract our perhaps natural and maybe even strong tendency towards having skewed perceptions of ourselves (on the other hand, we're ultimately the only ones with privileged access to our thoughts and will represent a black box to varying degrees to various people).

I don't know what argument Watts is making exactly here, but in general I'm quite skeptical of the notion that there's a true self somewhere hidden and unaccusable to us or that we're influenced or controlled by subconscious forces we're condemned to never fully understand or even be aware of (not that I'm ascribing this view to Watts).