I don't know if anyone has mentioned it here but giftrocket.com is also a discreet option. It goes on your credit card instead of being deducted from your bank account so you don't have to give them any information other than your card info and the recipient doesn't see that. They see only the name you set your account up with, which can be an alias.

I have only had one kicked back and that was because the giver put "session deposit" in the notes. They want you to at least pretend that it is a gift.

I agree that Paypal is the worst. They disclose private info on both sides of the transaction. Venmo is better but it is run by paypal and if you aren't paying attention to your settings you payments can be broadcasted to the Venmo community - which seems insane to Me.