In the past few months they have gotten more strict with this.
At Santander, at least, they have been requiring the proof of residency in the form of an official certified document similar to the one you get here from the police department. Spent about an hour in the bank discussing with the attendent and someone she was calling on the phone (at the central bank) trying to explain to them that certificate does not exist in the US. They did not have any examples or information on what folks from the US could use. I attempted to present a notarized copy of my mom's driver's license showing her address, along with her utility and bank statements, but they did not accept it.
If anyone figures out an acceptable form, please post!
I believe the limit of what you can send a month has been lowered from $5000USD to $1500USD. My understanding is that this is in total; not $1500USD per family member.
The kicker for me was that in addition, I was telling them I will be moving out of the country and inquired on how I can repatriate my account balance to my home country, and they told me that it's up to the central bank and I would have to make my case to them in person with documentation showing my employment ending in Argentina and beginning in the US, but that they have been denying folks in that situation anyway!