You hit the problem I've been dealing with on a project of mine. The short answer is yes, but.....
You see there are a couple of problems that have to be considered carefully. First, what the heck is going to power the circuitry you want to install. Second, do you want to know if there is power there or actually flowing. See, if you have power available and the bulb is burned out, the light isn't on. There are a couple of reasons for which one you want.
The power comes in on the common lead of the relay and is switched between the two carriers, so it matters where you pick up the power because in one position or the other, there won't be any power available on one or the other of the carriers. If you hook to the common line, you must be on the side where the power comes into the circuit. If you mix them up and get the side that leads to the light, you'll only have power to the circuitry when the light is on.
But these annoyances aside, there are two well known ways to tell if power is somewhere. First is to use something that samples voltage. People usually use a capacitor to feed an optoisolator hooked to the power line and then on the other side hook it up to an analog pin on the arduino. A little code and you can tell if there is mains power there. Second is to sample the current using something like an ACS712 to actually measure the power by hooking the output of it to an analog pin of the arduino. A little more code later you can tell if there is current flowing. A little (actually a lot) of searching on Google and you can turn up many examples of both of these ideas.
I don't have a definitive answer for either of these because my parts are on order to test the optoisolator idea. I have tested the ACS712 idea and it works really well to tell if there is current flowing, however the range of current is large and that can lead to troubles. For example, an incandescent light will pull around 800ma from my 110 supply, but a CFL light will pull less than a tenth of an amp. If I want to be able to tell current flow in a multiple light circuit as well as a single light circuit, it can get tough deciding what to do. This problem is compounded if you have parasitic devices (phone chargers, instant on TVs, etc.) on the circuit.
Not much help am I?