Rural Veterans Services Outreach and Training Act - Directs the Secretary of Veterans Affairs (VA) to carry out a program to make competitive grants to provide financial assistance to state departments of veterans affairs for the training of rural county veteran service officers in order to improve outreach and assistance to veterans, as well as their spouses, children, and parents, who may be eligible to receive veterans' or veterans-related benefits and who are residing in rural counties. Limits each grant to $2 million. Requires grants to be made on an annual basis.