Dogs definitely experience loss just like people do. Dogs are social animals and when their companion suddenly is not with them anymore they feel the void.
Dogs do not go through the exact same stages of grief as humans such as denial, anger, depression, bargaining, and finally acceptance. But they do experience denial, depression and acceptance.
If you had an at home euthanasia or if your other companion was present during the passing of your pet then the denial stage for dogs passes a litte quicker than if your other pet just leaves one day and does not return. It is difficult for dogs to comprehend if their companion just left temporarily or not. If they get to identify the deceased body I believe it hastens the denial phase.
Either way your dog may feel hopeful as though their companion will return, but after a couple days they start to realize the permanent loss and becomes depressed. This period of depression can vary. There are many factors that influence the length of grieving for dogs such as if there are other companions in the house, the depth of the bond, how long they were paired, etc. A period of depression is normal. It is fairly common for your grieving dog to stop eating, not participate in the things he/she usually enjoys doing, and pacing around the house. Sometimes you will wonder if your dog is grieving or if he is sick, since mental anguish can often mimick signs of physical illness in pets. If the reduced appetite persists more than a few days it is best to have your companion checked out by a veterinarian just to be on the safe side.
How long the grieving will take is unique to each individual. At some point, you and your pet will come to peace with the loss of your beloved companion. The memories will always be there. People or dogs never really move on, but we finally accept.