It is widely understood across all Buddhist traditions that the Buddha often spoke of persons and used words like 'I' and 'person' and 'self' and this is not seen as problematic or contradictory to the doctrine of anatman.
Just as it is said by the bhikkhuni Vajira:
“Just as, with an assemblage of parts,
The word ‘chariot’ is used,
So, when the aggregates exist,
There is the convention ‘a being.’
It is also self-evident that the Buddha would associate a proper name with individual beings and use this name conventionally over time to identify specific individuals. This is also not controversial for any Buddhist tradition that I'm aware.
However, there were times when "the self" was discussed in such a way that the Buddha would not answer such as the famous case of Vacchagotta. It is widely understood that the Buddha found Vacchagotta's questions or mindset about those questions as problematic. Specifically, he held that Vacchagotta was confused or had some invalid presuppositions about the self that directly violated or contradicted the doctrine of anatman.
Vacchagotta held to the notion that the self necessarily had hypostatic existence. Other words for this include "intrinsic", "inherent", "substantial", by different traditions of Buddhism.
There are some on this forum who hold that speaking of the self in this life - when it is merely for conventional communication - is fine and not in contradiction to anatman, but speaking of the self in other lives in the same continuity is strictly forbidden as always contradicting anatman and necessarily presupposing hypostatic existence.
Why insist that all discussions of future or past lives necessarily entails the presupposition of hypostatic existence? In short, why is it that hypostatic existence is only sometimes the basis for discussions in this life, but always the basis for discussions about past or future lives? Why do some hold so dearly that the Buddha could not be speaking of future lives in the very same conventional manner that the Buddha often used the word "I" to refer to himself in his present life... merely as a means of communicating the truth to worldly beings?
Why is it that some believe we can speak faultlessly of persons in this life and identifying them across various points in time in this life, but we are foreclosed of speaking faultlessly of persons in next lives or in past lives?
Update: trying to make this more clear given discussions in answers below...
Consider the Yamaka Sutta:
“What do you think, friend Yamaka, do you regard form, feeling, perception, volitional formations, and consciousness taken together as the Tathagata?”—“No, friend.” “What do you think, friend Yamaka, do you regard the Tathagata as one who is without form, without feeling, without perception, without volitional formations, without consciousness?”—“No, friend.”
“But, friend, when the Tathagata is not apprehended by you as real and actual here in this very life, is it fitting for you to declare: ‘As I understand the Dhamma taught by the Blessed One, a bhikkhu whose taints are destroyed is annihilated and perishes with the breakup of the body and does not exist after death’?”
From this we can draw four conclusions:
It is not appropriate to regard the Tathagata as possessing or consisting of the aggregates taken together
It is not appropriate to regard the Tathagata as not-possessing the aggregates or without them taken together
The Tathagata is not to be regarded as "real and actual" aka hypostatically existent
This is true both for future lives (the focus of Yamaka's question) as well as the present life
If this is so, then it was entirely appropriate to refer to the Tathagata as existing in his present life, but not to do so with the presupposition that the Tathagata was hypostatically existent aka 'real and actual', right? It was faultless to refer to the Tathagata as merely existing conventionally, right?
If this is so, then why is it a fault to refer to the Tathagata as merely existing conventionally in a future life? Indeed, Yamaka was rebuked for denying this very thing, right?!
With this question I'm interested in the perspective and answers from all the Buddhist traditions.