Using the Deep Space Network, NASA transmits a 20 kW radio signal from Earth. It takes almost 20 hours for the signal to reach the satelight Voyager 1 (so it is around 20 light-hours away from the Earth). The space probe’s sensitive antenna picks up the signal and replies using a 20-watt signal. It takes another 20 hours to reach to the Earth and as the signal travels through space, it weakens. By the time it reaches Earth, it’s barely detectable – but the DSN is able to detect it.
Theoretically, there isn’t really a limit on how far we can communicate with objects in space – as long as they reply back to us. With our current technology, we could reliably communicate with the Voyager 1 for thousands of years, even if it’s many light years away from us.