What happens in those few seconds if the coin was double spent? Are you notified before the next block comes in?
I'm not personally notified, no...
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.
