It sounds like you have a standard North American "Split Phase" electrical supply as supplied to most residential homes. As you mention: you have two "Line" connections plus a Neutral.
Most of your loads are 120 Vac and come from one of the Line conductors to Neutral. The remaining loads are 240 Vac and come from both of the Line conductors.
What you need to measure depends on what you are trying to measure. There are a couple of possibilities: you are trying to measure the current consumption of the entire household OR you want to measure the current consumed by only a single load.
If you are trying to measure the current consumed by the entire household, you have to measure the input current on BOTH of the Line conductors. This is because most loads return to Neutral but the remaining loads return to the other Line conductor.
If you are trying to measure the current consumed by a single device, you need to have current sensors on the total number of active power conductors (Line & Neutral) minus one.
In other words, if your load is strictly 240 Vac with NO Neutral connection, you need only ONE current sensor. OR: if your load is strictly 120 Vac with no connection to the other Line conductor, you also need only ONE current sensor.
However, if your load is something like an electric clothes dryer or electric range, you need TWO current sensors. That is because part of the load is between the two Line conductors, the remaining load is between either of the two Line conductors and Neutral.
Based on your revised question, I would say that you need only ONE current sensor. It doesn't matter which Line conductor the sensor is connected in series with.