When you sign the transaction, the wallet will generate you the partially signed bitcoin transaction (PSBT) which you will need to send to the second signer. Only what the second signer needs is the partially signed transaction (PBST) from you and then also sign it and broadcast the transaction. This is how multisig wallet works.
That helps. How do I send my partially signed transaction to the other person?
Also, if I originally had set up the wallets on Electrum, do I have to use Electrum for my future multi sig transactions? Or can I use Coinbase, Binance or any other exchange?