Durable Water Repellent (DWR) is the coating on the garment which allows the water to bead off the outer shell preventing the top layer of the jacket becoming waterlogged and diminishing breathability [see above]. DWR is not the main waterproof competent of the garment which is the membrane making up the mid layer of most jackets.
The easiest way to test your DWR is to pour a cup of water on the garment and watch. If the DWR is working effectively, then the water will bead off. If the DWR has degraded then the water will soak into the top layer of the fabric and wet out making the fabric look darker.
You will not need to reproof your garment with every wash. How regularly you will need to reproof your garment will depend on the quality of the DWR coating and the usage of the garment. For light usage, you should expect to reproof your garment 2 - 3 times per year for optimum repellency. If you use the garment for intense exercise (such as hiking, biking, running, climbing) you should expect to reproof much more frequently.