Connect both grounds together, use a digital input and if it's on, that is it?
It could be.
Will that cause any problems with the relay coil if connected from the active side to the relay, and the ground being common ?
No problem with the relay but it could be a problem for the monitoring arduino.
Is your relay 5V? Has it got a reverse polarity flyback diode across it?
If yes to both questions then it is safe to connect it directly to the arduino.
If not then you will have to buffer the input into your arduino through a transistor, potential divider, or opto isolator.