Ok reading this thread i thought that i should share my experience traveling 2 to 3 times a year for the last 5 years to Europe with my Argentine Passport. To get into Europe you need your argentine passport, that all the requirement, now if they see you suspicious they can ask you for health insurance and the hotel you are staying for at least the first 2 days (i never got asked but I know that this can happen). My recommendation is to just book a hostel for 1 or 2 nights, doesn’t matter how long you stay in Europe is just in the rare case they ask you and you can get a cheap insurance, normally many credit cards of Argentinean banks give free insurance for your travel in Europe so you can check that out, of course you have to be a client of that bank and have the credit card.
Now for what I have seen in airports custom in Spain, they were more tight in the past 5 to 10 years ago, now a days they rarely check anybody coming from Argentina unless the person seems very suspicious as they doubt anybody is going to work in there (there is no employment at all, especially on the younger people, and salary if you get a job there is not good anyway).
I always take my time to leave the plane as I hate to be pushing through people to get my things, I just stay sitting on the plane until the hall get free, so normally I am one of the last in the queue, I never saw a person getting stopped or detained in the airport, sure it can happen but mostly the whole passengers of the flight coming from Buenos Aires pass fairly fast through the customs.
Tip one, don’t go with invitations letters, they are of no use only make the official to start inquiring if you are staying at someone place that don’t help (the only cases I have heard of people having problem are when they present this letter, so avoid it, you don’t need it when travelling with your argentine passport), presenting this letter actually make things worse because then they can start thinking that maybe you have family there or a girlfriend and you are looking to stay, just stick to what I said before, get 2 night at a hostel for your arrival that´s all.
Tip two, Say hello with a smile in Spanish and you will be rarely even bothered with any further inquire, remember this only work if you have argentine passport, DNI is of no use in the Airports.
Travelling in brazil and rest of south America, I only travel with DNI inside south America (Brazil, Peru, Bolivia, Chile, not sure about Colombia and Ecuador) and that is the only legal requirement as argentine, now I don’t know what happen with DNI if you have a temporal residency in Argentina or if you are not Argentine, I never saw a DNI for foreigner so I don’t know if there is a special mark on it saying that you are not Argentinean.
Sorry if it got long, I hope this give some light on the matter. I just finish this month my dual Italian passport, so no more bothering for me, but to be fair it was never a problem to pass through Spain or any other country in Europe, and I found officers fairly friendly there, so I would say to not add all the custom officers in Spain in the same bag, I actually found customs there way more relaxed than when I arrive in Ezeiza.