In due time MoneyGram separated from the Western Union and became an independent brand. MoneyGram launched a new in-person bill payment service which will be delivered through agents in their domestic network, giving consumers an easy, low-cost method to pay their bills. What are the main benefits of money transfers by MoneyGram? Reliability - the international payment system gives complete assurance that the money will be paid to the recipient. Today the company MoneyGram International - an independent U.S. company, whose shares are traded on the New York Stock Exchange and MoneyGram money transfer is considered one of the largest international money transfer systems without opening a bank account.