The short answer is ‘no’, you do not need a coach.  That is to say that it is not a prerequisite or requirement to be a triathlete.  If you are completely new to the sport of triathlon (or to one of its three disciplines), some expert guidance will surely come in handy.  There are also people out there who need extra motivation to train on a daily basis, and coaches with training plans in hand can supply the extra incentive to get up and move.  There isn’t a triathlete out there who could not benefit from having a good coach (I emphasize the word ‘good’).  What you need to do is figure out what your goals are and what you would be willing to pay to get outside help to have a better chance at achieving those goals.  For many triathletes, the level of standard improvement they can expect to get from having a coach is not worth the price.  If you have an athletic background, know how to swim, bike, and run already, and you just want to participate, a coach is probably not for you.  If you have tried everything you know on your own and are a few minutes from the podium (or whatever your goal may be), a good coach can probably get you to where you want to be.