Using standard mastercard debit card in France & Italy

I think the ATMs in Italy work with mag strip cards also.
Although I can't confirm since all my ATM cards (Chase) are now chip cards and that's what I used this year.

Were they talking about using the debit card to make purchases or to get cash out of bank operated ATMs?

I can see there being issues using the no chip card to make purchases at some merchants. Many forgot or never knew anything about non-chip cards. The concept is alien to them and they don't realize there is a mag stripe reader on most credit card terminals.

If you use a bank operated ATM there also should not be any issues. They are mandated by the networks to support any type of card through 2018. If you try one of the other ATMs that are found in convenience stores and other similar shops and are not associated with a bank, those may not support mag stripes. Simple solution: use bank operated ATMs!

I have had no issues. Just back from London and got many Pounds from bank ATMs with my old non-chip card.

We were just in France (Normandy area and Paris) for 2 weeks in May and used our Mastercard debit to get cash many times with no issue. I do vaguely remember another post somewhere (maybe it was on Fodor's) about someone's BANK limiting their ability to use their MC, but as long as it's not your bank doing it, I don't think you'll have an issue.

I was worried about my non-chip card before our April Italy trip and begged my bank to issue a chip card, which they did when I mentioned traveling overseas. No problems whatsoever anywhere with the chip card. Several we talked with during our travels mentioned being unable to make purchases with their non-chip cards in Italy. We purchased our Euros before leaving the US and never tapped into the ATMs.

Regarding the option of purchasing Euro from the USA, I do not recommend that option.

The exchange rate and commissions you pay to buy currency in the US are pure extortion that borders racketeering.

I recommend you rely on your ATM card in Italy for the cash, and possibly use credit cards for larger payments (credit cards offer good exchange rates too).

If you buy currency in the US, limit it to the minimum necessary to cover you for your first day of arrival, in case you don't want to deal with finding an ATM at the airport (however at the Rome airport there are more than one, so that wouldn't be a problem).

I don't know if you have any near you but I did my currency exchange at TD Bank and got a good rate, $1.18 about two weeks ago.

The Euro was 1.125 - two weeks ago today. Your rate of 1.18 is a reasonable rate in the US - about 5% mark up. Sometimes it is much worse. Had I used my debit card in Italy that day, the exchange rate would have been about 1.133 to 1.138.

......We purchased our Euros before leaving the US and never tapped into the ATMs....... You can do that but that will be the most expensive way to obtain euro.

While your non-chip debit cards should work fine in ATMs, they won't work in "point of sale" machines such as train ticket machines. And, although I have never tried to use one in a store or restaurant, I understand they won't be accepted in that context either.