Transaction Confirmation between Users
The first confirmation is simply the fact that your transaction was included in a block. Any confirmations after that are simply new blocks that extend the chain on top of the history that includes the block with your tx.

To be notified about transaction confirmations, all one has to do is listen for the broadcasts of new blocks, and check if their tx is in it, or if the block extends the chain after their tx has been mined. If either of those hold true, the number of confirmations is incremented.

Due to this, it doesn't really matter who sent coins to whom. Every participant on the network is notified of which transactions are being confirmed, since they all receive new blocks.