Puppies Assisting Wounded Servicemembers Act of 2017 or the PAWS Act of 2017 This bill directs the Department of Veterans Affairs (VA) to carry out a five-year pilot program under which it awards grants to eligible nonprofit organizations to provide service dogs to veterans who suffer from post-traumatic stress disorder after completing other evidence-based treatment. An organization that receives a grant shall provide for each participating service dog and veteran: coverage by a commercially available veterinary health insurance policy, hardware clinically determined to be required by the dog to perform the tasks necessary to assist the veteran, payments for travel expenses to obtain the dog, and travel expenses required to obtain a replacement service dog. To be eligible for a grant, an organization must: (1) agree to cover all costs in excess of the grant amount to guarantee such benefits, (2) be certified by Assistance Dogs International, (3) provide one-on-one training for each service dog and recipient for 30 hours or more during a period of 90 days or more, and (4) provide an in-house residential facility or other accommodations nearby in which service dog recipients stay for a minimum of 10 days while receiving at least 30 hours of training. The VA shall develop metrics to measure the improvement in psychosocial function and therapeutic compliance and changes independence on prescription narcotics and psychotropic medication of veterans participating in the program. The Government Accountability Office must provide to Congress a briefing on the methodology established for, and a report on, the pilot program.