Most of the art you see comes from trained models that are either injected with Lora's (like a specific smaller training set) of a person, item, scenery, etc... Or trained by the user ontop of base 1.4/1.5 models. You can download all those from cites like Civtai (warning lots of NSFW), Or train them yourself using either online servers or with stablediffusion installed locally on your pc. It takes quite a beefy pc and more importantly a good GPU with lots of VRAM to run stablediffusion locally so I doubt there is anthing remotly good that you can run locally on your phone right now.

As you mentioned you can of course connect to online services, but I doubt you will get the freedom of having your own local instance.

Your best bet would be midjourney on discord, that should run on the discord phone app. Just keep in mind its a paying service, and has strong guidelines on what can and can not be generated.