You will hear it used mostly in fiction predominantly by working-class male characters (period dramas come to mind first) and when used in fiction it is often fairly clearly pronounced 「あちし」.
In real life, 「あちし」 is not pronounced clearly as in fiction. ItIt is more often howwhat 「あたし」 can sometimes sound like rather than how it is actually pronounced intententionallyintentionally. AnotherAnother way it can sound likebe perceived is 「あっし」. YouYou will hear these mostly around Tokyo and not really in western Japan.
I would say that the reason for the clearer pronunciation of 「あちし」 in fiction is its function as 「[役割語]{やくわりご}」("role language").
In real life as well, these variations of 「あたし」 is used by older male individuals around Tokyo. ThatThat 「あたし」 is used only by female speakers is just a rumour in the Japanese-as-a-foregnforeign-language world. DoDo not trust it.