Precision of estimating power consumption with a current sensor
To know power delivered to a load, you need current and voltage.

If your load has a unity power factor, for example a pure resistance, then you can assume the current is in phase with the voltage, and work from the current sensor output, and an estimate for what the RMS supply voltage is. If you have a nominal 230v supply, and you use that figure instead of the actual measured one, you could be up to 6% out, more in countries with a poor mains supply.

If your load power factor is not unity, then you must measure the instantaneous voltage as well as the instantaneous current. Multiplying the RMS values for voltage and current give you the VA of the load, which in the case of the resistor will be the same as the power. Multiplying the instantaneous values of voltage and current, and integrating them, will give you the true power, which is always equal to or less than the VA, never more.

As with a pure reactive load your true power will be zero, and the VA finite, the ideal resistor estimate will be 100% wrong.