Your first problem is that dog kidneys are roughly as efficient than human kidneys. That means that drinking dog urine is about on par with drinking human urine. IOW you will be drinking fluid with a concentration of salts and toxins that is already at or above the ability of your body to handle it. You would die in short order.
You could overcome that if you could somehow force feed the dog with large quantities of water to make it produce very dilute urine. All you need is a funnel, a length of hose and a very, very tolerant dog.
So, having solved that problem, you get to the next. The mammalian gut and kidneys do a really good job of filtering out microbes. So urine is effectively sterile and you have no need to worry about viruses or bacteria. So far so good.
What the kidneys are also very good at is filtering out non-living toxins such as benzene, heavy metals and many others. And by 'filtering out" I mean that they dump them into the urine for disposal. Because your dog will be using some of the water it drinks for cooling and faecal production, it will be urinating less water than it drinks, The result is that the urine will contain a greater concentration of some toxins than the water itself. IOW you will be consuming more of some toxins by drinking urine than by simply drinking the water.
You can mitigate this by force feeding the dog with water, but you will never be able to eliminate it, and in many cases you will never be able to improve the concentration above that of the water. When we say that kidneys filter out toxins, we mean that filter them out of the blood and concentrate them in the urine. The stuff you want to drink.
The we get to the third problem, which is that many toxins have their potency increased through ingestion. Once they get metabolised by the body they break own into even nastier compounds. Benzene, for example, breaks down into a dozen different toxins, many of which are much nastier than benzene itself. They also get rapidly excreted through the kidneys. And canine physiology is very different human. So the products that a dog’s body excretes may be highly toxic to humans and not something that your own body would produce at all if exposed to the progenitor directly.
As a result, the dog might be drinking water with 10 ppm of benzene, but excreting urine with 5ppm of phenol. That might sound like an improvement, except that phenol is about 6 times more poisonous than benzene. So as a result of drinking dog urine, you have effectively *doubled *the amount of poison you are drinking.
In short, your idea is acceptable if you are only concerned about pathogens or highly active toxins such as strychnine and provided you can force feed the dog. If you are concerned about salt or about relatively inactive toxins such as arsenic or hydrocarbons, you are outta luck. Your solution won’t help much, and unless you know exactly what the water is contaminated with you may be turning a relatively innocuous brew into a highly toxic one.