Yes, people do talk that way. What you are describing is not "reported speech". Rather, the speaker is simply shifting his time frame. For example, a man might say:
"Yesterday I was in the grocery store and I *SEE* this man stealing vegetables. I *AM* like 'whoa, you can't do that!' I *GO* to the store manager and *TELL* him what I saw. Then I left the store."
This is not the best of English, but it is common. The speaker is making a mental shift. He starts his sentence in the present moment. Then, INSIDE HIS BRAIN, he switches to the past. He starts speaking to you FROM THE PAST. Then, in his last sentence, he switches back to the present.
Don't ever talk this way in situations where accuracy is important. If the police officer asks you questions about the crime you witnessed, avoid colloquial language and use the correct tenses.