I definitely don't blame you for being skeptical and double checking to make sure you don't don't receive any surprise charges! I can assure you, the only way a carrier could charge you usage for using their service would be to have their SIM installed, and connected to a cellular network.
For example, if you arrived in Italy but kept your Verizon sim in the phone, had your cellular network connection setting off, even if your phone was on, you could still access all settings, camera, and connect to wifi around you to access the Internet or apps that connect thru wifi and Verizon wouldn't be able to charge you for any usage as the sim wouldn't connect to any cellular network. (You wouldn't have any signal strength bars showing, as there would be a no service indicator for cellist instead). It would operate similarly if you popped out the SIM card before arriving or before you power the phone on in Italy except you'd have a 'no sim' indicator showing instead of no service indicator.
You can test it out, even before your trip in the US by removing the sim, connecting to wifi, or even inserting a compatible sim from another domestic carrier like T-mobile or AT&T if you know someone that will let you test it out real quick to see what it looks like.
It's a 'Subscriber Identity Module' card that stores and transmits your Verizon phone number as well as security protocol to connect to available cellular networks, so with the network setting off or the card not installed the phone wouldn't be able to connect to any cellular network to be charged anything billing wise. If you were to put a friend's T-Mobile domestic sim in here or a TIM foreign sim while traveling and make a call, the phone number being transmitted would be your friends (here) or a local number from the city in Italy where you purchased the SIM/plan and would be billed based on the plan that is associated with the phone number. Also, only Verizon sims can connect to the Verizon network so you'd be safe.
Worst case scenario- if you travel to Italy and turn your phone on with the Verizon sim still installed before turning off your cellular network connection, this is the only way Verizon could charge you for international services.
It wouldn't be a surprise though, as as soon as your Verizon sim connected to an Italian cellular network without an international plan on your Verizon account you'd receive a free text message from Verizon saying- 'Welcome to Italy! Calls are $1.79/min texts are $0.50 to send & $0.05 to receive and data is $2.05/mb'. The message would also include the dialing pattern instructions to be able to make calls out. I wouldn't suggest trying this one out tho, as especially if your number and phone hasn't connected to a network in a while a bunch of text messages and emails could come in, and apps could sync and update causing unwanted charges to add up quickly. Verizon tends to roam on Vodafone in Italy, and while they have the best coverage and speed overall (especially in the populated areas) you can choose a SIM card and service from any compatible sim carrier that works specifically where you'll be traveling, and also based on the plan pricing they offer.
