With good instruction, you definitely CAN teach yourself how to sing incredibly well by learning without a "physical instructor" in the room with you.
In fact, many singers (including myself) make the most progress when practicing at home. I think it's a little easier when the pressure is off, and you can make as many funny noises as you like!
The key to teach yourself how to sing well, of course, is in the process that you follow.
You must use singing exercises that will improve your voice... AND you must understand why these exercises are important. In other words, you must understand how your voice works, and how to program your voice do function this way.
There are some incredibly bad exercises and you must avoid these, because they will not improve your singing at all.
The right singing exercises however will improve your voice more than you can imagine. Singing is actually not that complicated, and with the right process, you can get your voice functioning correctly for singing.
When you do this, I guarantee you won't believe the sweet sounds coming from your voice!
The best voice training available for home study purposes are on this page. The instruction is actually so good, that I would recommend using it before going to most instructors.
These instructions will firstly educate you as to exactly how your voice works, and then take you through exercises that get your voice functioning correctly for singing. You will actually notice a dramatic difference in the quality of your voice after the first hour of using the program.I kid you not...
Some things to remember when you teach yourself how to sing...