No you shouldn't have to worry about verification when depositing. No matter the method you deposit you still have other options to withdraw depending where you live. I deposited via Western Union and withdrew from Money Gram (at a Walmart). I didn't use my real name or address for the WU deposit and withdrew $500 per week for 4 weeks in a row showing only my id to check the name was me. As long as your info on the account is up to date then you don't have to worry about your ID at Money gram location, because they just want the name on the recipient to match your name on ID.
2 years ago I deposited a large amount via my debit card (checking account) and withdrew in the same method above by a check. If you have any issues use their live chat feature. They are not rude like many keep saying. They want to keep you and your business, not run you off.