I will butcher this so if any experts see this please don't flame me. I think you might be conflating ideas? You could definitely fine-tune existing embedding models or train your own from scratch but the goals of embeddings models are different than a LLM conversation. Embedding models are used for things like, classifying, search, image captioning...maybe at a high level anything where you have high dimensionality that you need to condense?

What you are asking for sounds like fine tuning an existing LLM...where the data will be tokenized but the outcomes are different? There is a lot of writeups on how people have done it. You should especially follow some of the work on Huggingface. To replicate talking to your friend though, you will need a very large dataset to train off of I would think and its unclear to me if you can just fine-tune it or you would need to train a model from scratch. So a dataset with 10s of thousands of examples and then you need to train it on a GPU.

https://www.anyscale.com/blog/fine-tuning-llama-2-a-comprehe...