The first thing to do is look for a real data sheet for the product from it's manufacturer.
**broken link removed**
Since they offer nothing substantial, you'd have to contact them to receive a good lie or measure a sample device.
From what I know of these, it would have a single LED with a reverse protection diode and a limiting resistor and would pass ~15mA on every half cycle so 120 * 0.015 / 2 = 900mW. You might want to be conservative and call that 1W for energy budget calculations.
The cost of operation on a monthly basis would depend on electric rates where installed but for example, if your rate were $0.25/kWh, it would cost $0.25 to run for 1000 hours. An average month has 730 hours. I'll leave the rest of the math to you.