Starting with my own (not Buddhist) analysis perhaps there are two types of "traditional/typical idea of a western self":
By "social" I include everything that would be impossible if one were (perhaps by some miracle of Science Fiction) alone in the world:
- Family and friends (e.g. "my family")
- Wealth, possessions (e.g. my house, my money)
- Social status (e.g. my job)
- Legal identity (e.g. my passport, my birthdate, my contracts)
- Reputation (e.g. my friends or enemies, my career)
I'll gloss over these, they aren't the main focus of this answer.
A lot of the suttas are by the Buddha and intended for monks who have "gone forth" and renounced these kinds of attachments and possessions and types of identity (also including "caste").
I think that Buddhism doesn't deny that such exist -- and I think it advises lay-people on how to fulfil some of their social obligations skilfully and wisely (e.g. Sigalovada Sutta (DN 31)) -- but that's not specifically "doctrine about self" (I'd consider it rather doctrine about "morality" etc.), and not the main focus of the suttas.
Still if you are interested in what advice can be found in the suttas specifically for lay-people, that's collected in this book titled The Buddha's Teachings on Prosperity: At Home, At Work, in the World.
Even so I think it's safe to say that, according to Buddhism, these things are all sankharas. They are (therefore) impermanent, not permanently satisfying, and their loss is an occasion for grief if one is unwisely attached to or clinging to them -- which is why (traditionally Meeting the Divine Messengers) the Buddha was inspired to "go forth" from his life as a lay-person.
By "personal" I mean to include what you may be aware of when you're alone in a forest -- as in fact a Bhikkhu might be or might have been.
Briefly, Buddhist doctrine categorises those as the "five aggregates".
Perhaps I'm wrong but I think the "form" includes the body (though there is another word for body, i.e. kāya) -- so you might say, "I have a body", or "I am a body", or "I die when the body dies", and so on.
Similarly for the other aggregates -- famously in the West, cogito ergo sum ("I think, therefore I am"), which I guess is a kind of identifying self with thought. There are also perceptions ("I see you"), feelings ("I am having a good time" or "I am sad").
The aggregates are also the "clinging aggregates" which are named at the end of the First Noble Truth (in SN 56.11).
After the Dhammacakkappavattana Sutta (SN 56.11) I think the next two suttas were the Fire Sermon and the Anatta-lakkhana Sutta, which introduces anatta which is the subject of (at least) nearly 200 topics/questions on this site.
In conclusion a good summary might be what MN 2 says about A thicket of wrong views.