Some signs to look for if you think your car's water pump is not working right, according to my experience.

Indicator #1. Grinding or Squeaking noises
A bearing that spins along the accessory drive belt is a key component of the car’s water pump. When the bearing in your water pump wears down, it makes a grinding or squeaking noise that can be heard even from inside the cabin. Many car owners report hearing ticking noises, while others report hearing growling noises.
To check if the noise is coming from your car’s water pump, open the engine bay while keeping the engine running and the auxiliary belt being driven. Keep a close eye on which accessory is making the noise. This is an early first indicator of a failing water pump.
When the accessory drive belt is in operation, a worn-out or damaged bearing might make some noise. This happens frequently after a lengthy period of running. However, if you just replaced the bearing, a loose auxiliary drive belt could be the source of the noise.

Indicator#2. Overheating
One of the most typical problems for car owners is overheating, which is particularly common among used vehicles. However, there are a number of conditions that might cause engine overheating on the road. A failing or malfunctioning water pump is one of them.
The main purpose of the water pump is for cooling down the car’s engine, which is why a faulty water pump can cause overheat to your engine. The water pump assists in the circulation of coolants from the coolant radiator to the heated operating engine to absorb the heat, then back to the coolant radiator to cool down.
The engine will become overheat without the water pump, causing other components of the engine to fail under high temperature. As a result, keep an eye on the engine’s temperature measurement in your car's instrument cluster. If the instrument needle is pointing well past the middle, you may have a defective water pump.
Due to its operation at high speed to maintain the transportation of the coolants at a necessary pace, bad water pumps are a typical problem.

 

These are EXACTLY the issues with my newly acquired 2003 Montero Sport LS. Guess I need to get the new water pump installed and not make those short trips to the grocery store risking greater damage/issues.