"Put in" is more commonly heard, but any of those would be understood. The only part that's idiomatic is "my two cents." You can use the expression with a variety of verbs or phrasal verbs.

“Put in” is most commonly used, but there’s no real difference.

Throw and put both have to be followed by “in” but no preposition is needed to give my two cents. The differences are very nuanced almost to the point of being insignificant. I would say give us the most common verb used with this expression. A person might also ask - what’s your two cents? Grammar is not always ironclad when it comes to idioms and popular expressions. ALSO - native speakers can take more liberties / make errors with idioms that are not noticed. Errors or liberties are flagged faster with non native speakers because the listener’s ear / mind is making more of an effort to understand already.

They are all referring to putting your own thoughts on the matter, your own. mind. All of verbs can be used - Throw my 2 cents at it, give my 2 cents on it or put my 2 cents in it.

I've also used "add my two cents" .