When you say "a local SIM card," you
do mean it was purchased in the same overseas location you are currently trying to use your phone + data, right? Or more specifically, it is a SIM card for a carrier who provides coverage where you are physically located? I think the answer to this is YES but I wanted to confirm.
Anyhoo, it sounds to me like you've done everything correct with Verizon to set this up. And because your SIM card is working for phone & text, and doesn't indicate you are roaming ... this tells me that your SIM card is set up properly and it is connecting to a carrier that is local to your current overseas location.
So probably the only thing missing is for you to call your SIM card provider and ask them to activate a data plan on your SIM card (and possibly having to wait 24 hours for it to take effect). Have you already done this?
If not, for some SIM cards, you can just send a text message to activate this, which is great if you don't speak the language well. And for all SIM cards, there are myriad data plans from which to pick. The best resource I've found for these kinds of dealings is the following page:
PrePaidGSM.net: National GSM Operators
Please keep us posted. I plan to travel to Italy (from the US) and will be in precisely the same boat, with precisely the same phone!
Good luck.