1999 Dodge Intrepid thermostat or water pump?

There's no way we can know the answers to any of your questions. You didn't bother to list the engine size. There's three possibilities and they're all very different. Also, we can't see the engine over a computer so you're the one who is going to have to look for the source of the leak. A leaking water pump is a good suspect but some are buried and can't be seen. You or your mechanic will have to make an educated guess based on where the coolant is seen running out. Thermostats don't leak, but the thermostat housing gasket can. That rarely occurs on its own. It's more likely to happen after the thermostat is replaced.

It's just as likely you have a leaking hose or a loose hose clamp. A pressure test will usually show that. If a leak is too small to see, you can add a small bottle of dark purple dye to the coolant, then search after a few miles of driving with a black light. The dye will show up as a bright yellow stain that you can follow back to the source.