Training is learning from the experience of OTHERS. Experience is learning from your OWN experience. Both have their place.
During training, your instructor (formal or otherwise) will take you on a dive to, say, 60'. After training you would ideally limit your depths to less than 60' for some time until you gain more experience & comfort, & then gradually increase your depths. Once you hit 60', you have reached the limits of your TRAINING. Now if you were to dive in familiar areas to 65', you might be exceeding your training, but not neccessarily your personal comfort zone. If five more feet of depth really bothers you, you should probably decrease by at least 20' & start all over. Somewhere along the way, a thinking diver will start worying about redundancy for their gas supply, but that is another discussion. Eventually you will end up with a diver doing 100'-120' dives that is quite capable of AND quite comfortable with doing these dives. Where did this diver exceed their training? At 60' 1". Where did this diver exceed their abilities? Never.
Moving on to the formal training thing. First let me say that I have some formal training, and I am working towards getting comfortable diving to the limits of my present certification. But suppose I, a certified cavern diver, were to meet someone who has been diving the caves of Florida for a considerable amount of time, is well known throughout the community as an excellent cave diver, has good people & communication skills, is willing to take the time & do the dives with me, etc & etc, what would make him or her less of a worthy instructor than any of the many highly respected instructors around this area? The fact that he is not authorized to give me a certification card from one of the major agencies? In fact, it is possible, if not probable, that this person might give me more time than a paid instructor would be able to, and therefore give me better instruction.
Now if you want to learn to cave dive inside of two weeks, look up someone other than my hypothetical mentor. If, however, you want to spend some time, there is no reason that this same hypothetical mentor can't teach you to perform cave dives as safely as possible. Just like, or perhaps better than, an instructor teaching for an agency. BTW, if you are my hypothetical mentor, PM me, I've been looking for you, LOL.