I am unsure about giving out contact information when completing transactions via money gram or at a bank -- is that a valid concern?
Cash deposits at a bank do not require that you provide any info. There have been isolated incidents where the teller asks for account info presuming the depositor has an account with the bank, but that isn't a requirement and they have no need to ask.
As far as MoneyGram asking they are probably wanting that in case you need to request a refund. There may be creative ways to fulfill their request without revealing your identity.
Anyway, blockchain has an option to deposit using SMS. I know it's a more expensive fee, but I feel like it's more anonymous than trying to make a cash deposit at a CVS or bank branch. Especially since you could presumably use a pre-paid phone that's funded with a pre-paid credit card?
I thought that was carrier billing not credit card / prepaid card.
It also is limited as to where that is available.
In the U.S., there are now several cash deposit methods that have no fee and require no fee.
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http://www.BitFloor.com (Deposit cash at Chase or Wells Fargo)
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http://www.BitMe.com (Deposit cash at Chase)
Bitcoins Direct accepts cash deposits at B of A, Wells Fargo and PNC Bank (minimum $500).