There are not any title police who enforce who can and cannot call themselves a historian, let alone a local historian. There are plenty of people with no degree who call themselves historians. Of course this is sketchy because they may know a lot of names and dates but they don't have a foundation in the methodology, they can't really practice the trade. I would expect a local historian to have a good knowledge of the overall history of a locality. They generally know a lot about the founding of cities and wartime involvement, but they should also know where to look for any other topic. So things like knowing all of the newspapers that have been published in that locality, etc.

You can absolutely do local history. It's very important, too. A lot of the basic primary source material is still available for research because somebody there on the ground happened to think it was valuable, happened to find some old ledgers that were about to be thrown out, volunteered at the local historical society and was there when the lady with an ancestor in the Civil War was trying to decide what to do with his old letters. And even if somebody wrote it up in some book in, say, 1929, it's not like the story of your community has been told: narratives of women, people of color, all tended to get too little attention, and new questions can always be asked. And there's nothing like working with old primary source material for things that happened right where you're sitting.

As an autodidact you will always be in danger of selective learning. You can stay ignorant about some things that a teacher would otherwise set you to study, and you can also easily fall into confirmation bias, reading things that are agreeable. So, if you were a Southern white male with lots of Confederates in your family tree, you might find it much more pleasant to read older books on dashing cavalry commanders like Forrest and Stuart, and avoid knowing much about the unpleasant slave society they were defending. Reading things that make you uncomfortable is part of the game.

If you're interested in insane asylums, you'd want to start by reading a decent general history of the treatment of the insane .It's not my field, but Colonial Williamsburg did do a nice little history of the insane asylum that used to be in the place of their Wallace Decorative Arts Museum: https://www.amazon.com/Quest-Cure-Hospital-Williamsburg-1773-1885/dp/0879351101 The author, Shomer Zwelling. also covered a little of the philosophies of treatment. The museum also did a short little video . ( Zwelling features in it as a patient....)

I would expect you could also post a question here, asking for suggestions for a good general history.

There was also a post here a while back with some useful references to papers on the deinstitutionalisation movement. The very thorough answer by u/amp1212 would suggest he/she might have some recommended reading as well.