The answer to this question is going to be yes and no. This depends on a couple of things. First, do you know what your Mission even is yet? That’s kind of important to know before you write a Mission statement. Sure you can write a corporate-style statement that talks about being a good person and being an agent of positive change and blah, blah, blah.
Those kinds of Mission statements are worthless for two reasons. First, they are meaningless tripe. There’s no description of a purpose or of action that you will take. It’s all feel good mumbo-jumbo. Second, they’re worthless, because that sort of thing should be taken for granted!
If you have to tell everyone how good you’ll be and well behaved and play nice with others, you’ve either been a really bad person before now or our society is so far gone that we’re all in serious trouble. Well, that may be true; but the fact that you will act like a good person should still be taken for granted by you at least.
Besides needing to know what your Mission is before you attempt to write one, you also need to ask whether or not you have internalized it already. This is a very similar argument to whether or not you should write down your Vision statement. If you are unsure of it, haven’t internalized it or have just decided on your Mission, then yes. Write a Mission statement.
If you are sure and have internalized it so that it permeates everything you do, then you have a legitimate option to not write it down. However, like the Vision statement, it’s better to do so. It gives you the opportunity to review it more thoroughly and serves as a better reminder and self-check mechanism to make sure you are on the right path.
Writing these down, like writing down your Goals, allows you to take a step back and examine them from a less emotional place. Examining these internally puts up barriers and strengthens your bias toward them. It’s harder to be honest because any criticism or change needed feels more personal. It’s harder to tell yourself you are wrong. It’s easier to cross out something and make some edits on paper or at the computer.
So, yes and no. No, it’s not absolutely necessary, but yes, it sure as heck is recommended.