Like T-Mobile, Google Fi doesn't charge for international texts. I have no idea what Verizon's policy is. I can continue to use my native telephone number to send and receive texts at no charge, even after activating an eSIM. That's actually the trickiest part of the process for me with a Samsung phone. I have to do some fiddling to get the phone to use my Fi eSIM for outbound texts; it seems to need that direction for each recipient rather than providing a single setting I can use. If I don't make that change, my outbound texts go nowhere, because there's no telephone number provided with a data-only eSIM.
Troubleshooting tips:
The eSIMs I've purchased from Airalo have come with good instructions, but the connection to the local phone provider isn't necessarily instantaneous. I assume there's an electronic handshake going on in the background that can take a few minutes on occasion. If you don't have cellular-data access within a few minutes of arrival in the new country and turning on your eSIM, first review the instructions provided by the eSIM provider to be sure you haven't skipped a step. Next, check to be sure roaming is turned on; I think I've had to do that manually for each eSIM. If still no joy, it won't hurt to reboot the telephone, though that's not usually required. That has seemed to help me on occasion, but those successes may have been purely coincidental.
I travel a lot in Europe, and I've learned not to freak out if I temporarily find myself without a data connection; I assume that's indicative of an issue with the local network I'm connecting to. That happened to me--usually briefly--when I was using T-Mobile, and it happens from time to time with Google Fi as well as occasionally when I'm using an eSIM.
Remember that there may be dead zones in rural areas, just as there are in the US.