Don't be concerned about what others think! As a runner, you deserve respect from other runners. Besides, runners love seeing others out on the roads or trails. We know how much enjoyment we get out of the sport, so why wouldn't we want to see other people doing the same thing? Also, remember that all runners were new to the sport at some point, so they can all relate to the struggles that beginners face.
If you're worried about what non-runners think, try not to get too hung up on that. Just remind yourself of all the great benefits that you're getting from running and they're missing out on. Be proud that you're doing something good for your physical and mental health.
Wearing the right clothes for running may make you feel more comfortable when running in public. For women, it's especially important to wear the right sports bra. If street harassment is a problem, try to ignore the person, hold your head high, and keep running. Yelling back or flipping him off will only exacerbate the situation.
You may also feel more confident about running if you are using the proper running form. Watch this video about proper running form to see how it's done. Even if your form isn't perfect, you'll feel better knowing that you're working on it.
You may also feel less self-conscious if you get a friend or family member to come along with you. An added bonus is that you can keep each other motivated to run.
Like anything else, the first time is usually the hardest. Once you've run in public a few times, you'll feel a lot more comfortable and be less concerned about other watching you.