Chess - Does one need formal training or not?

Do you need formal chess training to be more proficient or can you achieve high levels of chess just by studying on your own? 

I think this depends on your own personal goals.

It always helps to learn from other people's mistakes, and from their experience.

At a minimum, you have to play play against strong players (200-300 points higher than whatever your rating is), and review games from skilled players who are higher rated than you (like FM, IM, GM).

Sites like chess.com and chessgames.com give you a lot of what you need.

Can a person get a title without a coach? Of course. But it's not as easy. Some people will need a coach or mentor.

I grew up in Alaska, in a community without formal teachers, and got to 1900 by reading books, playing in many tournaments, and analyzing games with my opponents.  The top players in my club got to 2100 by the same means, and I believe GM Bryan Smith, also an Alaskan, got to quite a high level on his own.  So it is certainly possible.

As a personal observation, if you get to high level without a coach, you will tend to fight with your coach when you do have one....you are already set in your ways!