But if I have a Bitcoin client connected to the network then yes, it (along with all the other connected clients) should be sent every transaction shortly after they happen. It has to-- it might be generating blocks, and the whole point of block generation is to record the valid transactions the client has seen that haven't been included in a previous block.There's a discussion in this thread of how likely it is that an attacker could "split the network" to successfully double-spend coins.