Traders/Exchanges who require KYC are undermining the principles of cryptocurrency. Cryptocurrency is supposed to allow two parties to transact without actually needing to trust each other. The idea of “trust-less” transactions are put into practice via trust-worthy blockchain applications which are, or should be, open-source. You can trust the applications because you can inspect them and verify how they work.
LocalMonero, as an exchange, does not require KYC but for some reason some traders do ask for ID in some form. This is probably because they are advertising very large trades and want to ensure they are not going to be defrauded. Traders who are participating in smaller trades seldom ask for ID.
If you don’t want to share your ID, which you should never do, don’t trade with those people who do KYC. Eventually those people will not get any business and go away.