Yes, and as others pointed out below, this is essentially entanglement.
And you're describing entanglement in a misleading way. The most important part of entanglement is that you can't model it as two particles sharing secret data. When you touch one, you affect the output of the other. And this happens faster than light. It still looks random on each end, but once you combine the measurements you can see patterns.
My sense was that quantum physics basically makes sense if you go really small and have an understanding of how the particles interact. But entanglement is weird, because somehow there's also a mechanism for them to interact at distance?
(I know, I could try to look it up, but hopefully other people here have the same silly question)
As the GP said - to find out whether a particular change of state was random or a signal, one would need to compare readings from both.
Readings cannot be sent faster than the speed of light. Thus preserving causality.