Usually services like Western Union or MoneyGram only give the recipient the money, not the information about who and when sent it. But you can verify with them directly.
However, for legal/tax reasons, your friend might have to declare that it was a gift, and where it came from. So depending on the country of the destination you might not be able to completely "hide" from the recipient, even if the transfer service technically allows that.
In any case, when you transfer the money out from the US you'll have to provide your personal identification and information. Since the USA PATRIOT Act, it is impossible to transact "anonymously" (not sure if it ever was possible in the US, actually).