Updated: Oct 15, 2022
A core notion in the discourses of the Buddha concerns one's attitude towards experience, and in particular the ability of not appropriating it as 'my own.' The Buddha invites (e.g. in SN 22.59) to regard all the fundamental components of experience (the five aggregates) as 'this is not mine, this I am not' (anattā). Notice that in Pāli there is no devoted verb to mean 'to have,' hence possession and ownership is usually expressed using possessive pronouns ('mine') or the verb 'to be' plus a genitive (literally 'it is for me'). Fully grasping this point (non-ownership, and its experiential grammar) leads to full awakening. Why?
Crush-course on conditioned co-origination (cf. SN 12.10). The fundamental problem of existence is thirst (SN 56.11), the attitude of seeking for this or that state (or its annihilation). Thirst leads to appropriate its object (upādāna), and appropriation in turn endows both the object and the subject with an extra value, it makes the whole experience personal and precious, hence it gives it 'existence' (bhava). When there is appropriation towards something that exists, I also must exist (I appropriate, therefore I am). The becoming explicit of this owner of experience, namely 'myself,' is how I am born (jāti). In this sense, the sense of self is a product of thirst, its epiphenomenon. But like aversion and other habitual patterns, the condition of thirst is thirst itself, and hence this whole structure does not have a first point in time from which it originates. This means that there has always been a recycling between thirsting and self-ing. Because of thirst, a self is born, and because of a self, thirst is fostered, again, and again, and again.
This means that the complete cessation of thirst is what stops this recycling and constitutes final awakening. How does one get there? By countering thirst on two fronts. On the one hand, one counters the attitude that leads to appropriate contents and thus supports the sense of self and ownership (e.g. via sense restraint, morality, and other counter-attitudes aimed at diminishing greed and aversion, like friendliness). On the other hand, one counters the way in which the sense of self and ownership provides a justification, legitimization, and even encouragement for supporting thirst further and possibly indefinitely. By appreciating that this whole business of ownership seeks to obtain something impossible (control over what is uncertain, anicca), one can better see that any attitude of thirsting for possessing objects is doomed to fail since the start, and perhaps loose a bit of passion and infatuation for that. This is what counters ignorance (and by contrast, ignoring this impossibility is what fuels thirst).
Assuming that this outline is sufficiently intelligible, it is possible to use it to assess different interpretations of the Buddha's teaching on 'not-self,' by asking whether and to what extent they lead to undermining thirst or not. Two illustrations might help.
(1) 'The self is just a process.' This might be considered the underlying theory behind the Buddha's teaching of not-self. But it isn't. Sure, the five aggregates are processes, but what goes wrong with this view is that it does not undermine thirst at all. Anybody who ever liked a song or any other piece of music can very well realize that it is perfectly possible to crave and even be desperate for processes. Moreover, the view implicitly assumes that the teaching on 'not-self' has to be a sort of theory that defines what the self is or is not. But this in turns takes for granted the validity of the notion of 'existence' as something fundamental and essentially neutral. But it isn't. Existence is a sort of exclamation mark added to contents of experience for the sake of marking how special they are for me, particularly when they become mine. This is why both notions of existence and nonexistence must ultimately be rejected (SN 12.15). The teaching on not-self is not a theory about what the self is, but a practice about how to see (perceive) anything in experience as unsuitable for appropriating it.
(2) 'Not-self means becoming completely selfless and fully identify or merge somehow with others.' Perhaps emphasizing the way in which greed, aversion, and ignorance tend to create boundaries and distinctions, and how the notion of self relies on the same idea of differentiation from others, one might think that not-self means not-different, or even some form of transcendence beyond any subject-object distinction. But this is not the case. Fully identifying with others does not take thirst away, but simply shift its object. Instead of craving for what I want, now I crave for what others want. This surely annihilates my personal will and thirst, but it does not annihilate thirst (which has never been mine!). Some form of monasticism (Buddhist and Christian alike), as well as some lay movements and mass phenomena, put a strong emphasis on obedience as a way of submitting one's will to the will of another (Foucault has interesting things to say on this). But from the point of view of the discourses, this is just thirst for nonexistence (vibhava taṇhā). Empathetic identification with others might sometimes be helpful, and it is surely human, but it also has nothing to do with the teaching of not-self, since it does not entail any going beyond the ordinary structure of appropriation. And for what concerns the subject-object distinction, albeit in some traditions this is regarded as The Problem, this is not so in the discourses. The Buddha rather stresses the way in which consciousness delights in its objects (appropriates them) and recommends to relinquish that delight, by making consciousness unestablished (e.g. SN 22.55). But this does not mean that upon awakening (which is irreversible and occurs in this life, amidst the five aggregates), one will cease to perceive objects. In fact, he expressly declares (SN 35.232) that this wasn't the point. The point is the extinguishment of thirst with regard to the objects. The objects are fine, it is thirst that is not.