Is Electrum's SPV (thin-client) implementation not P2P (as opposed to Multibit's)?
MultiBit connects to (typically) four bitcoinds at random at start up. It then picks the best peer (using ping times, advertised blockchain height and the version of the peer to decide) and uses that as the download peer to get the block headers from.

It is thus using one peer to catch up the blockchain from where it previously knew about.

It listens to all four connections to hear about all the new broadcast transactions and uses this to work out a 'transaction confidence'. If four peers tell it a transaction exists, it has more confidence that the transaction is real.

When it sends a transaction, it sends it via the single download peer and then listens on the other peers to hear that transaction 'echo' back from the network. Once it has heard the transaction it just sent come back via other peer(s) it can be confident that the transaction is 'out in the Bitcoin network'.