BitPurse use a third party service provided by blockchain.info to retrieve public informations, like balance of an bitcoin address, transactions on an address, or emitting a transaction.
BitPurse store your wallet encrypted with AES CBC and can double encrypt your bitcoin private key. Your bitcoin private keys are never transmitted outside of your device expecting when you choose to use the export feature. In this case your wallet is exported encrypted.
Transactions are signed on your device and are relayed by blockchain.info