As I understand it, 'no separate self' and 'no self' are two very different things.
'No separate self' occurs through the recognition and abiding in one's true nature/Self beyond the personal identity/narrative self. In 'no separate self' the ever self-concerned, conflicted 'me' falls into the background and what takes its place is the unchanging equanimity, peace, spaciousness and clarity of the Self. This is a profound shift of identity: there is no sense of limitation or lack, and there is a deep felt sense of connectedness, belonging and unification not only to life but to the deeper reality underpinning it.
The Buddhists teachings would completely disagree with this. Anatta or 'non-self' (as opposed to 'no-self') in Buddhism states 'that no unchanging, permanent self or essence can be found in any phenomenon'.
'No self' as I understand it, is the complete absence of the narrative self or 'me', leaving only experiencing with no person at all at the center of it, no landing place, no reference point to an "I". It is a radical (and due to the experience of annihilation or 'ego death' not always welcome) shift into a stark, pristine, impersonal state of consciousness.