You can definitely watch your US based TV in Argentina using a Slingbox! Generally, if you are switching channels using the Slingbox, you are indeed going to change the channel at home.There are exceptions based on your set top box capabilities with dual tuners.
We have an article, When I change channels on my Slingbox, they also change at home that outlines this functionality.
Enjoy your Slingbox!
- Sling Support Team