When you turn the bathroom tap on the pump runs and pressurises the pipes to both taps, not just the bathroom one.
Sounds to me like the O-Ring seals have failed in your kitchen tap allowing the water to pass whatever position the tap is in.
Have you tried it all the way hot and all the way cold? Unless both the hot and vold water paths have faults then one of these positions will stop the water flowing.
Depending on the tap it might - or might not be repairable, only a close examination would tell. The one in our last abbey Aventura 330 was