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How to verify signature of multisig transaction (2-2) after it being signed by first owner? How can I understand if that owner used wrong private key => wrong signature?

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There is no way to check the transaction separately for each signature. Either the entire transaction is valid or it is invalid. When you receive the partially signed transaction, you would need to sign it with your own private key using the signrawtransaction command in bitcoind. It would give you two outputs (1) hex which is the serialized transaction and (2) complete. If the second output shows you "complete": true, then the transaction you received was signed with a valid private key. If it shows false then it wasn't.

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