What are the side effects of probiotics for dogs?
Probiotics are not a drug. Probiotics are referred to as "friendly bacteria" for a reason, side effects or negative reactions are extremely rare, however, like any supplement, they should not be taken or given to your dog without caution.
Since we're dealing with live organisms that communicate and operate inside the body, it is smart to know the basics about the individual species of probiotics, their activity, and how they affect the body of your dog. 
The most common side effect would be during the transitional phase when your dog is first introduced to probiotics. Your dog may experience excess gas or bloating. In some cases, your dog may appear more thirsty. These are all signs that your dog is detoxifying, and generally subside within a matter of days, or at worst, a couple weeks.*
With any diet change, you can expect your dog to experience some physical changes, but in the end, probiotics are a generally safe way to promote wellness long term in your dog.
Are there any risks with probiotics?
Each species of dog probiotics has unique characteristics and perform some specific function in the gut. Of the innumerable amounts of species of probiotics for dogs, some, such as E. faecium, are more controversial than others their ability to become pathogenic [1, 2, 3], not just to your dog, but to the human delivering the probiotics on the food.
Cites and References
de Perio MA, Yarnold PR, Warren J, et al.
Centers for Disease Control and Prevention (CDC). Vancomycin-resistant Staphylococcus aureus--New York, 2004. MMWR Morb Mortal Wkly Rep. 2004 Apr 23. 53(15):322-3.
Chang S, Sievert DM, Hageman JC, Boulton ML, Tenover FC, Downes FP, et al. Infection with vancomycin-resistant Staphylococcus aureus containing the vanA resistance gene. N Engl J Med. 2003 Apr 3. 348(14):1342-7./p>
- O'Sullivan GC, Kelly P, O'Halloran S, Collins C, Collins JK, Dunne C, et al. Probiotics: an emerging therapy. Curr Pharm Des 2005;11:3-10.
2. O'Sullivan GC. Probiotics. Br J Surg 2001;88:161-2.