No. Left and right when used to discuss placement on cars should be from the driving position. The oil filter is on the right front of the engine, the waterpump is on the left front of the engine. It doesn't matter where the water is dripping down. When my waterpump seals went, the drips were coming from the back of the oil pan. The water was running along the
top edge of the pan
Get the car on ramps and get under there with some towels. Keep drying up the drip and working back to the source.
If the belt was done less than 30K ago, I'd reuse it. Were the seals done at the same time?
How many miles are on the car?