2004 Jeep Liberty overheating

There's no ironclad way to test a water pump, if it's worn and not moving coolant you might feel a lack of pressure on the upper radiator hose. I've removed water pump with the vanes eroded down to nothing. You say you're losing coolant but don't see any contamination in the oil or signs of leaks, meaning you could be losing it out the tailpipe (due to a bad head gasket) or it could be running from the overflow bottle to the ground when engine is running (as coolant expands and isn't being pulled back into radiator it will eventually drain to the ground via the tube on the overflow bottle neck.) You could run a cooling system pressure test just to make sure there isn't a leak that has simply escaped your attention (like the bypass hose). Most parts stores can loan you the apparatus, they just ask for a deposit which they'll return once you're done with the tool. You can also rent or borrow a tool called Block Tester, which will test for combustion gases in the coolant, a sign of a leaking head gasket. Once you're certain there are no leaks or head gasket issues, then you can replace the water pump with reasonable confidence it's the culprit.

If you read the previously asked questions and answers related to Jeeps overheating, you'll see that the problem possibly stems from a faulty COOLING FAN MOTOR. I also heard "gurgling sounds" coming from my 2002 Jeep Liberty Sport 4x4. After informing my mechanic of my Jeep's "symptoms", he (correctly) asserted that the gurgling noise was from the radiator fluid/coolant literally BOILING inside the overflow container before it got so hot that the fluid spewed out (landing on the pavement below the front passenger seat).

Ken (previous responder) also mentioned the coolant overflow bottle, which supports the contents of my response.