The phrases you are asking about are more commonly spoken in British English than American English. They are more formal than using the 'to form'. 'One' is used as a gender neutral word meaning 'a person in general'. No one need feel left out. = No one needs to feel left out. (No person should feel excluded.) DO these sentences have the same meaning? Yes. You needn't go. = You don't need to go.

In modern spoken American English, we ALWAYS use "need to" and "don't need to." In written American English, we will always use "need to" in positive sentences, but will occasionally use "needn't" or "no one need [verb]" to give a formal tone. However, for Americans, this is rare. In British English, "needn't" and "don't need to" have slightly different meanings and usage. You can read about it in Zenildo's excellent link.

If you feel left out, you feel sad because you're not part of an activity. For example, some children are playing without Jane, so Jane feels left out. In your example, no one needs to feel left out, no one needs to feel sad for not being a part of an activity. Yes, "needn't" and "don't need to" have the same meaning in this sentence. We use these phrases to give permission to someone not to do something in the immediate future. For example: You needn't shout. I can hear you well. But if you're using need as a general necessity that's not related to the immediate future, don't use needn't, use only don't need to: You don't need to go to college to learn computer programming. You can study it at home. Sources: http://www.bbc.co.uk/worldservice/learningenglish/grammar/learnit/learnitv317.shtml https://dictionary.cambridge.org/us/dictionary/essential-american-english/be-left-out