After you clear customs/immigration and collect your bags, there is a Personal store in the Arrivals hall. My experiences with Personal have been consistently very bad, so I plan to go into the city and find a MoviStar store. MoviStar has more wifi bands feeding your cell than Personal or Claro, so more data in more places.
Last time I was in EZE (which was a couple of weeks ago) I logged onto the free wi-fi and was able to use the web without any difficulty.
I specifically tried to use the Skype app and it wouldn't work for me.
This means either
a) I haven't quite got the hang of this Skypey thing or
b) the ports that Skype uses might be blocked or
c) something else.
You can get both normal pre-paid SIM cards and eSIM (for iPhone) at the Personal shop in Ezeiza. You can recharge pre-paid cards for any provider at kiosks or using the Qiwi machines (find them here: https://qiwi.com.ar/ubicaciones.php). Personal is consistently the leader in the OpenSignal evaluations.
The free WiFi at Ezeiza is excelent, all over the arrivals hall, never had any problems with Skype, Whatsapp, email or anything else using it.
We arrived at EZE a week ago. The Personal store is in the arrivals hall, but on the second floor, outside security for departures. If you really want it. Last time we tried Personal they took our money and provided a SIM with no working voice or data service. When I went back to the retail store, a young female supermodel all but spit in my face and fussed with the phone until the network symbol showed, but the phone never provided any data or phone connectivity. So buyers beware.
We opted to try Movistar this time, visiting the corporate store on Av. Juan B. Justo between Palermo Hollywood and Palermo Soho. They provided me with 2 SIM cards, activated and registered on the network, and 2 local phone numbers. We were immediately able to call or text each other on our phones, and get internet connectivity. Then the fun began. To qualify for a one month tourist pack of 2GB and 100 min voice and unlimited WhatsApp, we needed to put AR$300 each through a kiosk. A helpful proprietor at the Kiosko just around the corner walked us through all that and even made change for a 500. We bought some munchies from him to reciprocate.
We both got cell messages that our AR$300 had been accepted and AR$100 had been added to our accounts successfully. The written instructions we got from the store said to text MES for month or SEM for week to a specific number, which we did for MES, and both got an acknowledgement plus another sentence in Spanish. This other statement turned out to be key - it was a requirement to "Activate" each phone via text message, and then to "Activate" pack for which we had just paid and qualified via the Movistar app.
Instead we assumed all was good and proceeded to surf and call and text with abandon. Later that day we both got warning messages that we had burned through 50mb of data and were being charged $AR16 for 50mb more. We thought it odd and shrugged it off. A couple of days later I was notified that my data speed was being reduced (and it was, essentially to zero) because I had burned through 100mb in a day and could get more at an increased rate of AR$35 for 50mb, but only if I authorized it via their app. Plus a bunch of messages that we had dropped below the AR$400 credits required to activate the tourist pack.
We opted to wait until our Spanish speaking local friend arrived for the weekend on Saturday. He showed us where we had ignored the steps and we proceeded to Authorize each phone. Each of us was required to submit 5 things via text messages to specific addresses:
1) Photo of passport first page
2) Selfie showing your face next to your passport
3) Photo of the card the chip came from zoomed in to show the numbers clearly and largely
4) Local address
5) Duration of stay
This went OK, despite being a bit of a hassle, with the only wrinkle that they texted my wife back to let her know that her SIM was registered to another foreigner. Since it was me, and she verified that, all was good.
Then we tried to download the app to activate the pack. There is no Movistar Argentina app available on the App Store, only Chile or Costa Rica or Spain or pretty much any other Latin country. We had my buddy call Movistar and discuss with them in Spanish. We were handed from rep to rep and none of them would believe that "no es possible" to download the app. Finally we got a rep via FB messenger (it was either that or Twitter - odd that WhatsApp was not an option), and that rep had us change the country codes in our phones from USA to Argentina, but even that didn't work. After spending over 2 hours trying to fix the problem, I texted back in Spanish (with my buddy's help) that we were going to walk away from Movistar and go to Claro, where the entire process could be handled in the retail store without all the extra steps. To this I received no reply for about 3 hours, and during dinner I got an apology that the rep was not trying to piss me off, but that the Movistar Argentina app was not available to anybody without a DNI. Even though this was their consistent instruction for their Tourist Prepago SIM on both website and paper and every other rep. He then said I could do it all through their website by logging into Mi Movistar.
I tried that today, and both accounts were easily accessed, and both clearly showed what balance we had chewed up by not having the pack and using the phones, and what was required to activate the Pack Prepago. A trip to the Kiosco down the street and I topped mine up, and came back to the apartment for the website, which showed the pack automatically activated for a month starting TODAY. Repeated for my wife's account, and now we are both good to go, with a 200mb (instead of 100mb) daily data limit before throttling.
Not a great experience, and it cost us AR$500 extra, but all good now and can be renewed in another month. I almost wonder if we are the first tourists to persevere through the process and actually get the advertised pack, because we could have just as easily kept topping off AR$500-600 per week for the rest of our stay and been fine (except for data throttling). I suspect that is what most tourists end up doing, and possibly by design. Even so, we had service from Day 1, so it has been a better deal than Personal despite the hassles.
The problem with the Movistar app is that if your iTunes store location is outside of Argentina (or maybe only outside of the Latin America region, not sure about that), you won't see it, even though it's actually there as the Movistar people said. You can confirm this by searching for 'Movistar Argentina app' in Google, when you find the "Mi Movistar Argentina" app and click on it, it will take you to the app store and show a "This app is currently not available in your country or region" message there.
Changing the country codes in the phones doesn't make a whole lot of sense, what you need to do if you really want that app is to change your country in iTunes. This may have other consequences, apps available in Europe or North America may not be available here, and any payment information would probably need to be updated.
Obviously YMMV, people's experiences vary, I've had no major problems with Personal, nor have the 4-5 people who've visited us here with activating a SIM card. The advantage of doing it in the airport is that they can take your passport details right there, but we've also done it in the Personal shop in Palermo. Your phone needs to be SIM-unlocked and have automatic network selection active as well.