Running with a dog is a great option if you're looking to get some more miles in but need a reliable companion that works around your schedule. Running with a human partner helps hold you accountable and they can conversate. But with humans, things come up. They can get busy, sick, or lose interest in running altogether- which leaves you to run all alone. But running with a dog is not only a safe option (especially if the dog is a bigger breed) but they will always be down to join you, so long as you get and keep them in running distance shape.
How far your particular dog can run, as well as how fast, depends on his or her specific breed. Bigger dogs that have longer legs and thus a longer stride are more likely to be able to keep up with your pace (some small dogs are capable as well but it will be harder on them to run as far). Also, be mindful of the environment you are running with your dog in. If you have a longer haired dog, it might not do as well in hot and humid conditions. And short haired dogs might not fare well in wind, rain, and cold temperatures. The terrain you are running over is an important factor to consider as well. Some breeds of dogs have more tender paws than other dogs, and rocky or wooded terrain (i.e. like those found on hiking trails) might be tougher terrains for your pup to conquer. One of the best parts about running with a dog is that they typically can run consecutive days in a row so if you are trying to train for a race, a dog can really help you meet your goal!
The biggest thing for you to consider is how old your puppy is. It is safest to wait until your puppy is full grown and its growth plates have closed. That way, you can ensure that your dog's growth is not impeded and your dog does not suffer lifelong injuries. Dogs are resilient, but like humans, they can get injured. Depending on your breed, most dogs will reach full growth after 1 to 2 years.