How does Stellar multi-signing happen practically? In Blog we read, that Government &amp; NGO came together to sign the transaction, and in docs we read that transaction is invalid if it includes signatures that aren't needed to authorize the transaction and that a transaction envelope wraps a transaction with a set of signatures. But how does the multi-signing process is intended to happen practically?

Let's say I make a transaction that required my and friend's signature, I sign it and publish it. Does it fail because of missing signature? Is my friend notified through stellar that there's a transaction that requires his signature, thus "inviting" him to take a look at it? Can my enemy find it and sign it to deny me contracts? And in case of 20 signers? And in case someone doesn't sign, when does transaction fails? Can he not-sign it in a way to fail the transaction before set expiration time?
The coordination of the tx envelope before it has the right number of signers happens by some other channel. If you and your friend both need to sign for any tx from an account, you would sign it and then email or IM the tx_blob to your friend. Your friend would then add their signature to the tx_blob and submit. 

If your enemy signs a tx you can simply remove your enemy's signature before submitting it to the network.  

If someone doesn't sign then the server you submit the tx to will reject it immediately.

Source: Multisig and Simple Contracts