Is there way to verify you own an NFT without actually knowing your token or whatever?

Sorry, I'm very new at this (never even bought a crypto) but I am curious about NFTs. Let's say I buy an NFT, is there a way for me to let someone verify that I own it.

Basically I am thinking about creating app or platform where people can display their NFTs but there is no risk of hacking or them losing their NFTs. Just a way to prove they actually own it. In fact, once they can prove they own it, I'll just generate a copy of their art to display.

Think of it like Instagram or whatever but the only pics you can display/upload are NFTs you own. Just trying to get an idea as to how difficult it is to create something like this that is secure.

Would such an app ever need to know a person's private key? Or would the public key suffice?

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Thanks!