How much power does a “smart meter” use?
The power used by the meter itself is called the burden.  The maximum value for both ANSI and IEC meters is set at a standard maximum of 5W for just the meter. Any communications or other auxiliary devices powered by the meter or directly via the mains are not included in this.  However, modern meters and communications together typically average well under 5W.

It's probably also worth noting that the measurement is done "downstream" of the device's own power supply so the energy that a meter measures does not include its own use.  The result is that individual customers don't directly pay for the cost of the electricity to run the meter.  Instead, it's part of the cost of doing business for the utility.