You do not need to register a service dog but you do need to have an illness that is considered disabling. So having a diagnosis for depression isn't enough, it needs to be a disability as well.

If you can train a dog yourself then you can train your own dog, but it isn't easy. I would recommend getting a trainer who specializes in training service dogs. You need a dog with solid temperament, so whether you get a puppy or a rescue, make sure not to get one because of looks. It can take months finding the right dog to do the job and not all puppies are service dog worthy. There are a ton of rejects in service dog organizations.

After all of that is taken care of and you have the dog, it will take a year or two before they are fully trained if they are an adult and more if it is a puppy. Going into public with your dog may sound fun, but there will always be people who try to cause problems.

Overall, having a service dog isn't something that is easy to have.

My dog isn't a service dog as I do not have a disability, but he does help me.

Doesn't leave my side during an episode.
Keeps me busy if my thoughts aren't healthy.
In stores that allow dogs, he buffers me and other people by standing behind me.
Because of my joints I sometimes have a hard time getting up from the floor so he will stand in front of me and I can use him to get up.