Best way to take advantage of dual-SIM or e-SIM smartphones while traveling?

One SIM for your European number, one for your regular U.S. number ?

We travel with two phones so mine gets the European SIM popped in to make local calls and texts while my husband keeps his with the regular AT&T number, in airplane mode, in case we need to make an emergency call home.

It's easy. With a dual sim phone, install your home cell number on the e-sim and leave the actual sim card slot empty. You will purchase a second SIM either on Amazon or when you get to Europe and then pop that in the SIM slot. Bingo, you now have two phone numbers, one which is your cell from back home and one which is your European cell number. Change the settings so that your cellular data plan is on the European SIM number.

When your cell number rings, you can talk just as you did at home. When the new cell number rings (doubtful, unless you know a bunch of folks in Europe) you can talk on that line. Basically, what you have is a phone with two lines.

Just be aware that not all cell companies support the e-sim. The major networks will support it but the discount companies will not. I plan on using T-Mobile for my overseas network and they support the e-sim while my primary company does not.

Here's my experience:
I have an ATT iPhone 11 with dual SIMs capability. I had my regular number switched to e-SIm at the ATT store.
I bought a Google Fi SIM (and monthly plan) and installed that in the empty SIM slot. I used my Google Voice number as that number, but you will be given a chance to choose a number or be given a number when you sign up.
Google Fi has decent European coverage and includes voice, data and texting and is included with the monthly fee. No extra international charges.
So when in Europe, I turn off the main ATT number (in the iPhone settings) and use only the Google Fi number.
Since I have iMessage that is tied both to my normal test number and my email address, I still get my iMessages.

When I first got my Google Fi SIM card, I wanted to test it out in the US, so I had both the Google Fi and the ATT lines active on my iPhone. Worked great. I got text and calls from both ATT and Google Fi on the phone.

Works great for me.

Thanks for sharing your experience with using these new phones with a plastic SIM slot and an e-SIM. It will be an added convenience when traveling.
I did a little searching and a basic aid to understanding is to note that the 'e' in e-SIM stands for 'embedded' not for 'electronic' as we might assume, so the e-SIM is the built-in identification chip and the plastic nano-sized SIM is one that you can swap physically.
I'm using a prepaid plan with the new iPhone SE so during this distancing period the carrier mailed me a SIM to stick in the slot myself and I (eventually) finished the port over of my number to this new phone without having to find an open brick-and-mortar store.

My T-Mobile service includes free overseas data roaming (2/3G) and text, and calls with charge. Here's my dual sim setup (android phone):

SIM1: T-Mobile SIM. No data, text and call only. The vast majority would be by either WhatsApp or WeChat. Both are free, popular in many countries, and preferred by the locals.

SIM2: Local SIM. Data only, for 4G/LTE speed. In some countries, the local SIMs either do not include text and call (Japan), or would charge extra for them.