gentlesoul certainly has a very good point about maintaining anonymity but I still feel OK using the web site. I suppose it depends on what you order and your tolerance for risk.
sasha44, could you have inadvertently chosen to use the "notify recipient" service when sending the WU transfer online? If you pay them to notify your recipient then of course they will need a phone number. This might be an easy mistake since WU tries to slide this service in, in kind of subtile way on their web site as a way for them to generate more fees.