My understanding was that birds did not get a shock because they do not complete a circuit. The birds are in contact only with the wire, so the electricity cannot flow through them. If they were touching the ground (or another object - tree, pylon, etc.) the wire would be earthed and electricity would flow through and kill the bird.
Apparently there are many cases when larger birds (vultures, storks, swans) have flown into wires and made contact with two separate wires - this creates a circuit and the bird receives a shock.
(I think this explanation is not actually different from cyanophile's response. The bird perched on a wire actually can create a circuit between the two feet but the voltage difference is very small and won't cause a shock, it is only when the bird is in contact with another object that it creates a voltage difference large enough to be felt.)
Are there any physicists who can comment?
